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PRESENTED

THE UNIVERSITY OF TORONTO

THE JOHNS HOP.KINS UNIVERSITY

BALTIMORE
1890

STUDIES IN LOGIC,

BY MEMBERS

OF THE

JOHNS HOPKINS UNIVEKSITY.

BOSTON:
LITTLE, BROWN, AND COMPANY.

1883.

UNIVERSITY PRESS:
JOHN WILSON AND SON, CAMBRIDGE.

PKEFACE.

THESE papers, the work of my students, have
been so instructive to me, that I have asked
and obtained permission to publish them in one
volume.

Two of them, the contributions of Miss Ladd
(now Mrs. Fabian Franklin) and of Mr. Mitchell,
present new developments of the logical algebra
of Boole. Miss Ladd's article may serve, for
those who are unacquainted with Boole's " Laws
of Thought," as an introduction to the most won
derful and fecund discovery of modern lo^ic.
The followers of Bo#le have altered their mas
ter's notation mainly in three respects.

1. A series of writers, — Jevons, in 1864;
Peirce, in 1867; Grassman, in 1872; Schroder,
in 1877 ; and McColl in 1877, — successively and
independently declared in favor of using the sign
of addition to unite different terms into one aggre
gate, whether they be mutually exclusive or not.
Thus, we now write

European + Eepublican,

to stand for all Europeans and Republicans taken

IV PREFACE.

together, without intending to count twice over
the European Republicans. Boole and Venn (his
sole living defender) would insist upon our writ
ing

European + Non-European Bepublican,

Non-Republican European + Bepublican.

The two new authors both side with the ma
jority in this respect.

2. Mr. McColl and I find it to be absolutely
necessary to add some new sign to express exist
ence ; for Boole's notation is only capable of
representing that some description of thing does
not exist, and cannot say that anything does exist.
Besides that, the sign of equality, used by Boole
in the desire to assimilate the algebra of logic to
that of number, really expresses, as De Morgan
showed forty years ago, a complex relation. To
say that

African — Negro

implies two things, that every African is a Negro,
and that every Negro is an African. For these
reasons, Mr. McColl and I make use of signs of
inclusion and of non-inclusion. Thus, I write

Griffin -< breathing fire

to mean that every griffin (if there be such a
creature) breathes fire; that is, no griffin not
breathing fire exists; and I write

Animal -< Aquatic,

to mean that some animals are not aquatic, or

PREFACE. V

that a non-aquatic animal does exist. Mr. McColl's
notation is not essentially different.

Miss Ladd and Mr. Mitchell also use two signs
expressive of simple relations involving existence
and non-existence; but in their choice of these
relations they diverge both from McColl and me,
and from one another. In fact, of the eight sim
ple relations of terms signalized by De Morgan,
Mr. McColl and I have chosen two, Miss Ladd
two others, Mr. Mitchell a fifth and sixth. The
logical world is thus in a situation to weigh the
advantages and disadvantages of the different
systems.

3. The third important modification of Boole's
original notation consists in the introduction of
new signs, so as to adapt it to the expression of
relative terms. This branch of logic which has
been studied by Leslie Ellis, De Morgan, Jo
seph John Murphy, Alexander MacFarlane, and
myself, presents a rich and new field for investi
gation. A part of Mr. Mitchell's paper touches
this subject in an exceedingly interesting way.

The method of using the Boolian calculus —
already greatly simplified by Schroder and by
McColl — receives still further improvements at
the hands both of Miss Ladd and Mr. Mitchell,
and it is surprising to see with what facility their
methods yield solutions of problems more intri
cate and difficult than any that have hitherto been
proposed.

VI PEEFACE.

The volume contains two other papers relating
to deductive logic. In one of these Mr. Grilman
develops those rules for the combination of rela
tive numbers of which the general principles of
probabilities are special cases. In the other, Dr.
Marquand shows how a counting machine, on a
binary system of numeration, will exhibit De
Morgan's eight modes of universal syllogism.

There are, besides, two papers upon inductive
logic. In the first, Dr. Marquand explains the
deeply interesting views of the Epicureans, known
to us mainly through the work of Philodemus,
irepl <T77/xeiW KOLI crrjjjLeictxTewv, which exists in a
fragmentary state in a Herculaneum papyrus.

The other paper is one which, at the desire of
my students, I have contributed to the collection.
It contains a statement of what appears to me to
be the true theory of the inductive process, and
the correct maxims for the performance of it.
I hope that the thoughts that a long study has
suggested to me may be found not altogether
useless to those who occupy themselves with the
application of this kind of reasoning.

I have to thank the Trustees of the Johns
Hopkins University, for a very liberal contribu
tion toward the expenses of this publication.

C. S. PEIKCE.
BALTIMORE, Dec. 12, 1882.

CONTENTS.

PAGB

THE LOGIC OP THE EPICUREANS 1

By Allan Marguand.

A MACHINE FOR PRODUCING SYLLOGISTIC VARIATIONS 12

By Allan Marquand.

NOTE ON AN EIGHT-TERM LOGICAL MACHINE ... 16
ON THE ALGEBRA OF LOGIC 17

By Christine Ladd.

ON A NEW ALGEBRA OF LOGIC 72

By 0. H. Mitchell.

OPERATIONS IN EELATIVE NUMBER WITH APPLICA
TIONS TO THE THEORY OF PROBABILITIES . . . 107
By B. I. Oilman.

A THEORY OF PROBABLE INFERENCE 126

NOTE A 182

NOTE B 187

By C. S. Peirce.

THE LOGIC OF THE EPICUREANS.

BY ALLAN MARQUAND.

WHEN we think of the Epicureans we picture a friendly
brotherhood in a garden, soothing eacli other's fears, and
seeking to realize a life of undisturbed peace and happi
ness. It was easy, and to their opponents it became
natural, to suppose that the Epicureans did not concern
themselves with logic ; and if we expect to find in their
writings a highly developed formal logic, as that of the
Stoics, our search will be in vain. But if we examine
the letters of Epicurus, the poem of Lucretius, and the
treatise of Philodemus1 with a view to discovering the
Epicurean mode of thought, we find a logic which out
weighs in value that of their Stoic rivals. This logic is
interesting to us, not only because it is the key to that
school of Greek Philosophy which outlasted every other,
but because a similar logic controls a powerful school
of English thought.

The logic of Epicurus, like that of J. S. Mill, in op
position to conceptualism, attempts to place philosophy
upon an empirical basis. Words with Epicurus are signs
of things, and not, as with the Stoics, of our ideas of

1 Gomperz: Herkulanische Studien I. Leipzig, 1865. Bahnsch: Des
Epicureers Philodemus Schrift Ilepi o-wduv nai ffweidxreuv. Eine Darleg-
ung ihres Gedankengehalts. Lyck, 1879.

2 THE LOGIC OF THE EPICUKEANS.

things.1 There are, therefore, two methods of inquiry :
One seeks for the meanings of words ; the other, for a
knowledge of things. The former is regarded as a pre
liminary process ; the latter, the only true arid necessary
way of reaching a philosophy of the universe.

All our knowledge is to be brought to the test of
sensation, pre-notion, and feeling.2 By these we do not
understand three ultimate sources of knowledge. De-
mocritus 3 held to only one source, viz., Feeling ; and Epi
curus, who inherited his system, implicitly does the same.
But each of these modes of feeling has its distinguishing
characteristic, and may be used to test the validity of our
knowledge. It is the peculiarity of sensation to reveal to
us the external world. Sensation4 reasons not, remem
bers not ; it adds nothing, it subtracts nothing. What
it gives is a simple, self-evident, and true account of
the external world. Its testimony is beyond criticism.
Error arises after the data of sensation become involved
in the operations of intellect. If we should compare this
first test of truth with Hume's " impressions," the second
test, pre-notion, would correspond with Hume's " ideas."
Pre-notions 5 were copies of sensations in a generalized or
typical form, arising from a repetition of similar sensa
tions. Thus the belief in the gods6 was referred to the
clear pre-notions of them. Single effluxes from such re
fined beings could have no effect upon the senses, but
repeated effluxes from deities sufficiently similar produce
in our minds the general notion of a god.7 In the same

1 The hypothesis of XeKrd, or of immaterial notions, was a conceptu-
alistic inconsistency on the part of the Stoics. The Epicureans and the
more consistent empiricists among the Stoics rejected them. -Sextus
Empiricus, Math. viii. 258.

2 Diogenes Laertius, x. 31. 8 Sextus: Math., vii. 140.
* D. L., x. 31. 6 D. L., x. 33. 6 D. L., x. 123, 124.

' Cicero: De Nat. Deor., i. 49; D. L,. x. 139.

THE LOGIC OF THE EPICUREANS. 3

manner, but through the senses, the continued observa
tion of horses or oxen produce in us general notions, to
which we may refer a doubt concerning the nature of the
animal that moves before us.

The third criterion, Feeling (in the limited sense), was
the ultimate test for ethical maxims. The elementary
forms are the feeling of pleasure and the feeling of pain.
A fourth criterion was added, viz., The Imaginative rep
resentations of the intellect. Its use is by no means
clear.

Upon this foundation rises the structure of Epicurean
logic. When we leave the clear evidence of sense we
pass into the region of opinion, away from the stronghold
of truth to the region where error is ever struggling for
the mastery of our minds. A true opinion l is character
ized as one for which there is evidence in favor or none
against ; a false opinion, one for which there is no evi
dence in favor or some against. The processes by which
we pass to the more general and complex forms of know
ledge are four : Observation, Analogy, Resemblance, Syn
thesis.2 By Observation, we come into contact with the
data of the senses ; by Analogy, we may not only enlarge
and diminish our perceptions, as we do in conceiving a
Cyclops or a Pygmy, but also extend to the unperceived
the attributes of our perceptions, as we do in assigning
properties to atoms, the soul, and the gods ; by Resem
blance, we know the appearance of Socrates from having
seen his statue ; by Synthesis, we combine sensations, as
when we conceive of a Centaur.

As a matter of fact, Epicurus regards only two proces
ses, — Observation and Analogy. Our knowledge, then,

1 D. L., x. 34, 51. Sextus: Math., vii. 211.

2 D. L., x. 32. The Stoics held a similar view; see D. L., vii. 52.

4 THE LOGIC OF THE EPICUREANS.

consists of two parts : l (1) The observed, or phenomena
clear and distinct to consciousness ; and (2) The unob
served,2 consisting of phenomena which are yet to be ob
served, and of hidden causes which lie forever beyond
our observation. The function of logic3 consists in in
ference from the observed to the unobserved. This was
called a sien-iiiference. According to Epicurus there are
two methods4 of making such an inference; one resulting
in a single explanation, the other in many explanations.
The former may be illustrated by the argument,— Motion
is a sign of a void. Here the void is regarded as the
only explanation to be given of motion. In other cases
many explanations are found equally in harmony with
our experience. All celestial phenomena belong to this
class. That explanation which alone represents the true
cause of such a phenomenon being unknown, we must
be content to admit many explanations as equally prob
able. Thus thunder5 is explained by supposing either
that winds are whirling in the cavities of the clouds, or
that some great fire is crackling as it is fanned by the
winds, or that the clouds are being torn asunder or are
rubbing against each other as they become crystallized.
In thus connecting celestial and terrestrial phenomena,
Epicurus aimed only to exclude supernaturalistic expla
nations. This done, he was satisfied.

In the garden at Athens this logic took root and grew ;
and by the time that Cicero visited Greece and sat at the
feet of Zeno,6 he may have listened to that great repre-

1 Philodemus: Rhet,, lib. iv., i. col. xix.

2 That is, TO Trpocr^vov Kai TO &St]\ov, D. L., x. 38.

8 D. L., x. 32. 66 ev /ecu irepl TWV ddyXuv diro T&V <f>au>0(J,frwv Xp?j <T77/xei-
ovcrOai.

* Ibid., x. 86, 87.

5 Ibid., x. 100. Cf. Lucretius, lib. vi. 95-158.

6 See Zeller's Stoics, Epicureans, and Sceptics. London, 1880, p. 412,
n. 3.

THE LOGIC OF THE EPICUKEANS. 5

sentative of the Epicurean School discussing such ques
tions1 as, — How may we pass from the known to the
unknown ? Must we examine every instance before
we make an induction ? Must the phenomenon taken
as a sign be identical with the thing signified ? Or, if
differences be admitted, upon what grounds may an in
ductive inference be made ? And, Are we not always
liable to be thwarted by the existence of exceptional
cases ? — But such questions had no interest for Cicero.
He was too much an orator and rhetorician to recognize
the force of the . Epicurean opposition to dialectic. The
Epicurean logic 2 to him was barren and empty. It made
little of definition ; it said nothing of division ; it erected
no syllogistic forms ; it did not direct us how to solve
fallacies and detect ambiguities. And how many have
been the historians of philosophy who have assigned
almost a blank page to Epicurean logic !

With a supreme confidence in the truth of sensation
and the validity of induction the Epicureans stood in con
flict with the other schools of Greek philosophy. The
Stoics, treating all affirmation from the standpoint of the
hypothetical proposition, acknowledged the validity of
such inductions only as could be submitted to the modus
tottens. The Sceptics denied the validity of induction
altogether. Induction was treated as a sign-inference,
and a controversy appears to have arisen concerning
the nature of signs, as well as concerning the mode
and validity of the inference. The Stoics divided signs
into suggestive and indicative? By means of a sug
gestive sign we recall some previously associated fact :
as from smoke we infer fire. By indicative signs we
infer something otherwise unknown : thus motions of

1 Philodemus TTC/H <yrnj.etC)v, col. xix.-xx. 2 Cicero: De Fin., i. 7, 22.
3 See PraiuUl's Ges. d. Log., i. 458.

6 THE LOGIC OF THE EPICUKEANS.

,the body are signs of the soul. Objectively a sign was
viewed as the antecedent of a valid conditional propo
sition, implying a consequent. Subjectively, it was a
thought, mediating in some way between things on the
one hand, and names and propositions on the other.
The. Epicureans looked upon a sign as a phenomenon^
from whose characters we might infer the characters of
other phenomena under conditions of existence suf
ficiently similar. The sign was to them an object of
sense. In considering the variety of signs, the Epicureans
appear to have admitted three kinds ; but only two are
defined in the treatise of Philodemus.1 A general sign is
described as a phenomenon which can exist whether the
thing signified exists or not, or has a particular character
or not. A particular sign is a phenomenon which can
exist only on the condition that the thing signified act
ually exists. The relation between sign and thing sig
nified in the former case is resemblance ; in the latter, it
is invariable sequence or causality. The Stoics, in devel
oping the sign-inference, inquired, How may we pass from
the antecedent to the consequent of a conditional prop
osition ? They replied, A true sign exists only when
both antecedent and consequent are true.2 As a test,
we should be able to contrapose the proposition, and see
that from the negative of the consequent the negative of
the antecedent followed. Only those propositions which
admitted of contraposition were allowed to be treated as
hypothetical.3

On this prepositional ground, therefore, the Epicurean
must meet his opponent. This he does by observing
that general propositions are obtained neither by contra
position nor by syllogism, nor in any other way than

1 Philod., loc. cit., col. xiv. 2 Sextus : Math., viii. 256.
3 Cicero: De Fato, 6, 12; 8, 15.

THE LOGIC OF THE EPICUKEANS. 7

by induction.1 The contraposed forms, being general
propositions, rest also on induction. Hence, if the in
ductive mode of reasoning be uncertain, the same degree
of uncertainty attaches to propositions in the contra-
posed form.2 The Stoics, therefore, in neglecting in
duction, were accused of surrendering the vouchers by
which alone their generalizations could be established.3
In like manner they were accused of hasty generalization,
of inaccurate reasoning, of adopting myths, of being rhet
oricians rather than investigators of Nature. Into the
truth of these accusations we need not inquire. It is
enough that they cleared the way for the Epicureans to
set up a theory of induction.

The first question which Zeno sought to answer was,
" Is it necessary that we should examine every case of
a phenomenon, or only a certain number of cases ? " 4
Stoics and Sceptics answered, The former is impossi
ble, and the latter leaves induction insecure. But Zeno
replied : " It is neither necessary to take into considera
tion every phenomenon in our experience, nor a few cases
at random ; but taking many and various phenomena of
the same general kind, and having obtained, both from
our observation and that of others, the properties that are
common to each individual, from these cases may we
pass to the rest." 5 Instances taken from a class and
exhibiting some invariable properties are made the basis
of the inductive inference. A certain amount of variation
in the properties is not excluded. Thus from the fact that
the men in our region of country are short-lived, we may
not infer that the inhabitants of Mt. Athos are short
lived also ; for " men in our experience are seen to vary
considerably in respect to length or brevity of life."

1 Philod., loc. cit., col. xvii. 2 Ibid., col. ix.

3 Ibid., col. xxix. * Ibid., col. xix. 13-15.

5 Ibid., col. xx. 30-col. xxi. 3. 6 Ibid., col. xvii. 18-22.

THE LOGIC OF THE EPICUREANS.

Within limits, then, we may allow for variation due to the
influence of .climate, food, and other physical conditions ;
but our inference should not greatly exceed the limits of
our experience. But, in spite of variations, there are
properties which in our experience are universal. Men
are found to be liable to disease and old age and death ;
they die when their heads are cut off, or their hearts
extracted; they cannot pass through solid bodies. By
induction we infer that these characteristics belong to
men wherever they may be found, and it is absurd to
speak of men under similar conditions as not susceptible
to disease or death, or as having the ability to pass through
iron as we pass through the air.1

The Epicurean looks out upon Nature as already di
vided and subdivided into classes, each class being closely
related to other classes. The inductive inference proceeds
from class to class, not in a hap-hazard way, but from one
class to that which resembles it most closely.2 In case the
classes are identical, there is no distinction of known and
unknown ; and hence, properly speaking, no inductive in
ference.3 In case the classes are widely different, the
inference is insecure. But within a certain range of re
semblance we may rely as confidently upon an inductive
inference as we do upon the evidence of sense.4

In speaking of the common or essential characters, the
basis of induction, it was usual to connect them with the
subject of discourse by the words ?;, Ka06, or nrapo. These
words may be taken in four senses : 5 (1) The properties
may be regarded as necessary consequences ; so we may
say of a man that he is necessarily corporeal and liable
to disease and death. (2) Or as essential to the concep
tion or definition of the subject. This is what is con-

1 Philod., loc. cit., col. xxi.

2 Ibid., col. xviii. 20 ; col. xxviii. 25-29. 3 Ibid., col. vi. 8-10.

4 Ibid., Frag. 2, 5-6. 6 Ibid>j col> xxxiii> 33_col> xxxiv§ 34

THE LOGIC OF THE EPICUREANS. 9

veyed in the expression, " Body as body has weight and
resistance ; man as man is a rational animal." (3) That
certain properties are always concomitant. (4) The
fourth sense, lost in the lacunas, appears from the fol
lowing examples to involve degree or proportion : " The
sword cuts as it has been sharpened ; atoms are im
perishable in so far as they are perfect ; bodies gravitate
in proportion to their weight."

Zeno's theory of induction may be formulated in the
following Canons : —

CANON I. — If we examine many and various instances
of a phenomenon, and find some character common to
them all, and no instance appears to the contrary, this
character may be transferred to other unexamined in
dividuals of the same class, and even to other closely
related classes.

CANON II. — If in our experience a given character is
found to vary, a corresponding amount of variation may
be inferred to exist beyond our experience.

The most important objection made to this theory was,
that phenomena exist in our experience exhibiting pecu
liar and exceptional characters, and that other exceptions
might exist beyond our experience to vitiate any induc
tion we may make. The following examples are given : l
The loadstone has the peculiar property of attracting iron ;
amber, of attracting bran ; the square number 4 X 4, of
having its perimeter equal to its area. Exceptional char
acters are found in the Alexandrian anvil-headed dwarf,
the Epidaurian hermaphrodite, the Cretan giant, the
pygmies in Achoris. The sun and moon also are unique ;
so are time and the soul. Admitting such exceptional
phenomena, the Epicurean replies, that the belief that a
similar state of things exists beyond our experience can

1 Philod., loc. cit., col. i., ii.

10 THE LOGIC OF THE EPICUREANS.

be justified only inductively.1 And exceptional phenom
ena must be viewed not as closely resembling, but as
being widely different from, other phenomena. Induc
tions concerning loadstones must be confined to load
stones, and not extended to other kinds of stones. Each
class of exceptional phenomena offered a new field for
induction, and hence could be said to strengthen and not
to weaken the inductive argument.2

The correctness of all inductions could be tested by
the rule of Epicurus for the truth of opinion in general.
An induction is true, when all known instances are in its
favor, or none against; it is false, when no instances are
in its favor, or some against. When the instances are
partly one way and partly another, we cannot reach
universal conclusions, but only such as are probable.3

This theory of induction was completed by a considera
tion of fallacies, summarized in a work called the " Deme-
triac." 4 These consisted in —

1. Failing to see in what cases contraposition is ap
plicable.

2. Failing to see that we should make inductions not
in a hap-hazard way, but from properties which resemble
each other very closely.

3. Failing to see that exceptional phenomena are in
no way at variance with the inductive inference, but on
the other hand add to its force.

4. Failing to observe that we infer from the known to
the unknown, only when all the evidence is in favor and
no shadow of evidence appears to the contrary.

5. The failure to perceive that general propositions
are derived not by contraposition, but by induction.

When we compare the work of Zeno with that of

1 Philod., loc. cit., col. xxv. 2 Ibid., col. xxiv. 10-col. xxv. 2.
8 Ibid., col. xxv. 31-34. 4 Ibid., col. xxviii. 13-col. xxix. 24.

THE LOGIC OF THE EPICUREANS. 11

Epicurus, an important logical difference is brought to
view. Both are occupied with the sign-inference, and
look upon inference as proceeding from the known to the
unknown. Epicurus, however, sought only by means of
hypothesis to explain special phenomena of Nature. Zeno
investigated generalizations from experience, with a view
to discovering the validity of extending them beyond our
experience. This resulted in a theory of induction, which,
so far as we know, Epicurus did not possess. In the
system of Aristotle, induction was viewed through the
forms of syllogism, and its empirical foundation was not
held in view. The Epicureans, therefore, were as much
opposed to the Aristotelian induction, as they were to the
Aristotelian syllogism. It was Zeno the Epicurean who
made the first attempt to justify the validity of induction.
The record of this attempt will give the treatise of Philo-
demus a permanent value in the history of inductive
logic.

It is refreshing to see the formalistic and rhetorical

atmosphere which had surrounded the subject of logic
breaking away, and an honest attempt being made to
justify the premises of syllogism. As yet, this had not
been done by all the moods of the philosophers.

It is also interesting to find in the ancient world a
theory of induction which rests upon observation, sug
gests experiment, assumes the uniformity of Nature, and
allows for the variation of characters.

A MACHINE FOR PRODUCING SYLLOGISTIC
VARIATIONS.

BY ALLAN MARQUAND.

FROM any syllogism a number of logical variations
may be derived. One operation by which this may be
accomplished is contraposition. This operation consists
in effecting a change in the order of the terms of a
proposition, the state of things which the proposition is
designed to express being supposed to remain unchanged.
Thus the state of things expressed by the proposition
" every A is a B " may be expressed also by " every
non-B is a non-A," or by the form, " there is a B for
every A."

We proceed now to apply this principle to the syllo
gism. For our notation let us take letters A, B, C,
etc. for general terms, and express their negatives by
writing dashes over them, A, B, C, etc. Let a short
curved mark over a letter indicate that its logical quan
tity has been changed ; thus, A, B, C, etc. A general
term will be thus made particular, and a term already
particular will be made general. Let us use the sign
-< for the copula.1 We may then express the syllo
gism Barbara in the form

A-<B

B-< C

.-. A -< C

1 This notation is that used by Mr. C. S. Peirce, "On the Logic of
Relatives." Memoirs Am. Acad. of Arts and Sciences, vol. ix, 1870.

A MACHINE FOR SYLLOGISTIC VARIATIONS. 13

From this as a starting-point we may produce formal
variations by various modes of contraposition. The ex
hibition of two such forms will suffice.

(1) We may regard the logical quality of the terms
and contrapose. The form A -< B then becomes
B-< A, or, "every non-B is a non-A."

(2) We may regard the logical quantity of the terms
and contrapose. The form A -< B then becomes
B -< A. The latter form we may take to mean, " there
is a B for every A," or " the B's include all the A's."

Applying these two kinds of contraposition to Barbara,
we obtain the following variations : —

Qualitative Variations.

B<A A-<B B-<A A<B B-<A A-<B B-<A
Fundamental B^CC 0<B 0-<B B-<C B<C CKB C-<B

Form' /. A-<C .'. A<C .-.A-<C .'.C-<A .-.C<A .-. C -< A .-. C -< A

J Quantitative Variations.

B-<C

B-<A AKB B^A A-<B
B-<0 C-<B C<B B^C

.-. C-<A /.

These may be classed as two figures according as the
conclusion has the fundamental or contraposed form ; or
they may be classed as four figures according as one or
other, or both, or neither premise has been contraposed ;
or as eight figures, if we regard merely the relative posi
tion of the terms. The number of such variations may
be indefinitely increased by admitting other modes of
contraposition, or by starting from other syllogistic
forms. All these variations may be easily produced
by a mechanical contrivance. In order to secure this
I have constructed a machine (Fig. 1) which pre
sents to view three flaps in which are inserted cards
containing the premises and conclusion of the syllogism
which is to undergo transformation. Each flap, on

14 A MACHINE FOR SYLLOGISTIC VARIATIONS.

making a half-revolution, presents its proposition in a
contraposed form. The flaps terminate on one side of

FIG. l.

Scale i in.

the machine in one-inch brass friction wheels. These
are marked a, 5, and c in Fig. 2. The wheels d, e,
and / are, respectively, one, two, and four inches in
diameter. Upon each of these wheels is fitted the sec
tor of a wheel of like dimensions. Wheel d has on its
outer side a sector of 180° ; wheel e, on its inner side,
one of 90° ; wheel /, on its outer side, one of 45°. The
friction of these sectors against the wheels a, 5, and c
causes the half-revolutions of the three flaps. By turn-

A MACHINE FOR SYLLOGISTIC VARIATIONS. 15

FIG. 2.

ing a crank attached to wheel d, the proposition A -< B
is contraposed at the end of every turn, B -< C at every
alternate turn, and A -< C at the end of every fourth
turn. Eight turns of
the crank will exhibit
seven variations, and
restore the fundamen
tal syllogism to view.
This mechanism
could be readily ex
tended so as to pro
duce variations in a
Sorites. A Sorites of
n propositions would
require, to contrapose
its conclusion, a wheel
of 2n~1 inches in di
ameter. We should
secure, as in the syl
logism, 2n — 1 varia
tions for each kind
of contraposition.

Scale J in.

NOTE. — The Syllogistic Variation Machine will unfold to view
the combinations of three logical terms and their negatives ; or if we
take the letters B — C, A — U, D — T, we obtain the words

BCBCBCBC

AAUUAAUU
DDDDTTTT

NOTE ON AN EIGHT-TERM LOGICAL MACHINE.

I HAVE completed the design of an 8-term Logical
Machine, of which a 4-term model is now nearly fin
ished. If the premises be reduced to the form of the
combinations to be excluded, as suggested by Boole and
carried out by Venn, the operation of excluding these
combinations may be performed mechanically by this
machine. I have followed Jevons in making use of keys,
but require for the 8-term machine only eight positive
and eight negative letter keys and two operation keys.
The excluded combinations are exhibited by indicators,
which fall in the squares of one of my logical diagrams
(Phil. Mag. ON. '81) from the perpendicular to a hori
zontal position. The non-excluded combinations, which
constitute the conclusion, are exhibited by the indicators
which are left standing.

ON THE ALGEBRA OF LOGIC.

BY CHRISTINE LADD.

THERE are in existence five algebras of logic, — those
of Boole, Jevons, Schroder, McColl, and Peirce, — of
which the later ones are all modifications, more or less
slight, of that of Boole. I propose to add one more to
the number. It will bear more resemblance to that of
Schroder than to any of the others; but it will differ
from that in making use of a copula, and also in the
form of expressing the conclusion.1

ON IDENTICAL PROPOSITIONS.

The propositions which logic considers are of two
kinds, — those which affirm the identity of subject and
predicate, and those which do not. Algebras of logic
may be classified according to the way in which they
express propositions that are not identities. Identical
propositions have the same expression in all. Of the
logical theorems which are identities, I shall give those
which are essential to the subject, and for the most part
without proof.

(1) The sign — is the sign 'of equality, a = 5, a
equals #, means that in any logical expression a can

1 The substance of this paper was read at a meeting of the Metaphysical
Club of the Johns Hopkins University, held in January, 1881.

18 ON THE ALGEBRA OF LOGIC.

be substituted for 6, or b for #, without change of value.
It is equivalent to the two propositions, " there is no a
which is not 5," and, " there is no b which is not a."

(2) The negative of a term or a proposition or a
symbol is indicated by a line drawn over it. a = what
is not a.

(3°) a + b = what is either
a or b. As a class, it takes in
the whole of a together with
the whole of b, what is com
mon to both being counted
once only. It has the quality
of either a or b, and hence
the quality of the entire class
is the quality common to a
and b. The only qualities pos-

(3') aXb = what is both a
and b. As a class, it is what
is common to the classes a and
b. As a quality, it is the
combination of all the quali
ties of a with all the qualities
of b. When relative terms
(XXI)1 are excluded from
consideration, ab may be writ
ten for a X b.

sessed by every member of the
class " lawyers and bankers '* are the qualities which lawyers
-and bankers have in common.

When arithmetical multiplication and addition are to
be considered at the same time, logical multiplication
and addition may be indicated by enclosing + and X in
circles. The addition of logic has small connection
with the addition of mathematics, and the multiplication
has no connection at all with the process whose name it
has taken. The object in borrowing the words and the
signs is to utilize the familiarity which one has already
acquired with processes which obey somewhat similar
laws. There would not be the slightest difficulty in
inverting the operations, and expressing logical multi
plication in terms of addition, and logical addition in
terms of multiplication. The essential processes of sym
bolic logic are either addition or multiplication (for
greater convenience, both are used), and negation. The
1 References in Roman numerals are to the titles at the end.

ON THE ALGEBRA OF LOGIC. 19

latter process renders any inverse processes which might
correspond to subtraction and division quite unneces
sary, and it is only on account of a supposed resemb
lance between the logical and the mathematical processes
that an attempt to introduce them has been made.

(4;) aaa = a.

(5r) abc = bca = cba.
(6')

(4°) a + a + . . . = a.

(5°) a+b+c=b+c+a=c+b+a.

(6°) a + be=(a + b)(a + c).
The symbol oo represents the universe of discourse.
(Wundt, Peirce.) It may be the universe of conceivable
things, or of actual things, or any limited portion of
either. It may include non-Euclidian w-dimensional
space, or it may be limited to the surface of the earth,
or to the field of a microscope. It may exclude things
and be restricted to qualities, or it may be made co
extensive with fictions of any kind. In any proposition
of formal logic, oo represents wha,t is logically possible ;
in a material proposition it represents what exists.
(Peirce.) The symbol 0 is the negative of the sym
bol oo ; it denotes either what is logically impossible, or
what is non-existent in an actual universe of any degree
of limitation.

(7')
(80 a =

(9')ao=a+oo = a+(H5)+...
(100 ab + al) + ab + aB

(7°) ' a + a = oo.

(8°) a=a+0 = a+bt+cc+

(9°) Q = aQ =

(10°)

The first member of this equa
tion is called the complete development of two terms. The
complete development of n terms, (a + a) (b + 1) (c + c) . . . ,
consists of the sum of 2" combinations of n terms each.

(11') a+ab + abc+ ... = a \ (11°) a(a+1>)(a+b+c). . .=a.
This is called by Schroder the law of absorption.

20 ON THE ALGEBRA OF LOGIC.

The only process which presents any difficulty in this
calculus is the process of getting the negative of a com
plex expression ; and that difficulty is very slight if the
right method is selected. There are three different
methods, of which the last is of most frequent use.
The first proceeds from the consideration that ab + al +
db + ab is a complete universe (10'), and that what is
not one portion of a universe must be some other portion,
if it exists at all. It follows that

ab — ah + ab + ab,

(12) ab + al = ab + «5,

ab + ab + ab = abj

and the process is the same for the complete develop
ment of any number of terms. This is the only rule
made use of by Boole and by Mr. Jevons for obtaining a
negative. If certain combinations of ten terms are
given as excluded, to get those which are not excluded
it is necessary, by this method, to examine 1,024 combi
nations of ten terms each.

The second method is contained in the following
formulas : —

(13')

ab — a + b.

(13°) a + b = al.
a + I = ab.

That is, the negative of a product is the sum of the
negatives of the terms, and the negative of a sum is the
product of the negatives of the terms.1 For example.

1 Professor Wundt (XVIII., p. 257, note) makes the singular mistake
of supposing that because x (y -\- 2) = xy -\- xz, the parentheses must be
removed before performing any general operation upon an expression.
The negative of a product of the form (a -f- & -f- <?) m, he says, is not
dbc-{-m, but (a-f-m) (b-\-m) (c-}-m); and in working his problems he
actually expresses it in this way, performs the indicated multiplication,
obtaining doc -f- (a-f- 5 -f- c) m-\-m, and then reduces this expression by
the absorption law (11') to aoc-f- ra.

ON THE ALGEBRA OF LOGIC. 21

a + be + def= a (5 + c) (a + e +/).

This rule was first given by De Morgan (" On the
Syllogism," No. III., 1858). It may be proved in the
following way : —

by (12),

a + b = a (b + b) + (a + a) b
= ~ab~-

ab — ab + al> + aft + aB
= a (b + 1) + (a + a) I

by (12).

It appears that with the use of the negative sign the
sum and the product are not both essential to complete
expression. A sum can be expressed as the negative of
a product, or a product can be expressed as the negative
of a sum. The dualism which has been pointed out by
Schroder, and which he indicates by printing his theo
rems in parallel columns, is, then, not an essential quality
of things, but merely an accident of language. We prefer
to say " what is either black or blue," to saying " what
is not at the same time both not black and not blue ; " but
one is as easy to express symbolically as the other. It
would not be difficult to develop the whole subject in terms
of multiplication alone, or of addition alone ; but the gain
in simplicity is not equal to the loss in naturalness.

The third method of obtaining the negative of an ex
pression is by means of the following equation : —

(14) pab + qal + rob + sab — pab + qab + fab + sab.

That is, consider any number of the letters as the
elements of a complete development (10'), and take the
negative of their coefficients. The reason is the same
as for (12), — the two expressions together make up a
complete universe, since

pab + pa b = ab, etc.
It is necessary to observe that if any part of the develop-

22 ON THE ALGEBRA OF LOGIC.

ment is wanting, its coefficient is 0, and the negative of
its coefficient is QO . For instance,

O + q + r) xy + stxy + uvwxy
=pqrxy + (§ + t) xy + (u + v + w) xy + xy.

The entire number of combinations excluded by the
first member is 7.25 + 26 + 25, and that included by
the second member is 25 + 3.26 + 7.26 + 28, and together
they make up 1024. This rule is given by Schroder only
(XIV., p. 19). It is much easier of application than
(12) or (13), except when the given expression bears no
resemblance to a complete development.

(15) An expression may be said to be in its simplest
form when it is represented by the smallest possible
number of letters. It does not follow that it is then in
its least redundant form. For instance, in

a + b, = a + dbj = dL + b,

a + b is simpler than either of the other expressions,
but it is redundant. It is

a (b + 1) + (a + a) b,

which contains the combination ab twice ; while
a + abj = a (b + I) + ab,

contains each combination once only. The reduction
of an expression to its simplest form may usually be
accomplished by inspection. Take, for example, the
expression

a + be + abd + add.
We have

a+ a($ + c)d= a + bed,
and

be + bed = be + d.

Hence the whole expression is

a + be + d.

ON THE ALGEBRA OF LOGIC. 23

If the reduction is not evident, it may be facilitated by
taking the negative of the expression, reducing it, and
then restoring it to the positive form (XVI., vol. x.
p. 18).

OX THE COPULA.

I shall adopt the convention by which particular pro
positions are taken as implying the existence of their
subjects, and universal propositions as not implying the
existence of their subjects. Mr. Jevons would infer that
the two propositions

The sea-serpent is not found in the water,
The sea-serpent is not found out of the water,

are contradictory ; but Mr. McColl, Mr. Venn, and Mr.
Peirce would infer that the sea-serpent does not exist.
With this convention, contradiction can never exist
between universal propositions nor between particular
propositions taken by themselves. A universal propo
sition can be contradicted only by a particular propo
sition, and a particular only by a universal. The above
premises are inconsistent with

The sea-serpent has (at least once) been found.

With this convention, hypothetical and categorical pro
positions receive the same formal treatment. If «, then
b = all a is b — a implies b. (Peirce.)

Algebras of Logic may be divided into two classes, ac
cording as they assign the expression of the " quantity "
of propositions to the copula or to the subject. Algebras
of the latter class have been developed with one copula
only, — the sign of equality ; for an algebra of the
former class two copulas are necessary,1 — one universal

1 Every algebra of logic requires two copulas, one to express propo
sitions of non-existence, the other to express propositions of existence.
This necessarily follows from Kant's discussion of the nature of the affir
mation of existence in the " Critik der reinen Vernunft." — 0. S. Peirce.

ON THE ALGEBEA OF LOGIC.

and one particular. The following are the propositional
forms which have been used by the principal recent
Avriters on the algebra of logic : 1 —

Traditional.

Boole
and
Schroder.

Jevons
and
Grassmann.

Grassmann.

McColl.

Peirce.

Uni
versal.

All a is b

No a is b

a=vb
a=vl)

a = ab
a = aB

a + b=b
a +1=1

a : b

a : 1}

a-<b
a-<l

Partic
ular.

Some a is b
Some a is not &

va = vb
va = vfi

ca = cab
ca=cab

ca-{-b = b
ca-{-b=:b

a-f-5

a^-b

a^<l
a^<b

v is a special symbol, used to denote an arbitrary,
indefinite class. It is immediately eliminated from the
universal propositions, which then appear in the forms
ab = 0, ab = 0, respectively. In particular propositions
" v is not quite arbitrary, and therefore must not be eli
minated" (III., p. 124). Jevons makes no distinction
between an indefinite class symbol, c, and any other
class symbol. With Mr. McColl, every letter denotes a
statement. By a : b is meant that the statement that
any object is a implies the statement that it is also 5;
but this does not affect the working of the algebra.
The negative copula, -f-, is the denial of the affirmative
copula, : , arid a -f- b', or, as he also writes it, (a : 5')', is
read " a does not imply non-5." Mr. Peirce' s symbol for
the same copula is a modification of ^. a ^< b is the
denial of a -< 6, and is read, " a is not wholly contained
under 6." a and b may be either terms or propositions.
The copula -< has an advantage over : in that it ex
presses an unsymmetrical relation by an unsymmetrical

1 Mr. Venn has collected some two dozen ways in which "a is b" has
been put into logical form.

ON THE ALGEBRA OF LOGIC. 25

symbol, a -< b may be written b >- a and read, " b
contains a."

This quantified copula (-< or :) is positive for uni
versal propositions, and negative for particular proposi
tions. Another kind of quantified copula is possible,—
namely, one which is particular when positive, and uni
versal when negative. Instead of writing

( A^< B

\ and \ A is-not-wholly B
(or A is-partly-not B,

we might write

AVB I an(l J AV B

A is-wliolly-not B ) (A is-partly B,

and it will appear that this latter plan has certain advan
tages. It comes perhaps a little nearer to common use.
The sense " wholly " is usually attached to both is and
is-not, but somewhat more strongly to the latter than to
the former. We say, for instance, " flowers are fra
grant," meaning that flowers are nearly always fra
grant ; but " leaves are not blue " means that leaves
are absolutely never blue. " Knives are sharp " would
be taken as true ; " knives are not blunt " would excite
opposition in the mind of the hearer.

The sign V is a wedge, sign of exclusion. A V B is to
be read " A is-not B," or " A is excluded from B" The
sign V is an incompleted wedge, sign of incomplete ex
clusion. A V B is to be read " A is in part B" or
u A is not-wholly excluded from B" V is made into V
by the addition of the negative sign ; what is not not
wholly excluded from anything is wholly excluded from
it. AvB and AvB are contradictory propositions ;
each simply denies the other.

The eight propositions of De Morgan are then, —

26 ON THE ALGEBRA OF LOGIC.

A V B A is-not B ; no A is B.

A V B A is in part B ; some A is .#.

^ V B A is-not not-£ ; all A is ^.

A V B A is partly not-1? ; some A is not .B.

AVB What is not ^4 is-not B ; .4 includes all B.

V ^ What is not A is in part B; A does not include all B.

What is not A is-not not-^; there is nothing he-
sides A and B.

What is not ^ is in part not-B ; there is something
besides A and B, —

where V connects terms that, exist, while V connects
terms which may be non-existent. Only six of these
propositions are distinct, since there is no difference of
form ^between A\/S and A\/£, nor between A V 5?
and A V B.

Propositions expressed with the copula : or -< are
called inclusions ; propositions expressed with the cop
ula V may be called exclusions. Exclusions with an
even number of negative signs are positive (affirmative)
propositions ; those with an odd number are negative
propositions (De Morgan, " Syllabus of a Proposed Sys
tem of Logic," p. 22). But the distinction, as Professor
Wundt and others have pointed out, is unimportant. The
only division of propositions which is of consequence is
the division into universal and particular. The copulas
V and V are intransitive copulas, — a kind of copula of
which De Morgan proposed to investigate the characters
(" Syllabus," p. 31). They are symmetrical copulas, and
the propositions A V B, A V B, may be read either for
ward or backward. It is from the fact that there is no
formal difference between subject and predicate that the
advantages of this algebra follow. There is, however, a
slight difference in meaning between A\/B and B V A ;

ON THE ALGEBKA OF LOGIC. 27

the subject of the proposition is more evidently the subject
of discourse. The propositions, " no men are mortal/'
and " there are no mortal men," convey the same infor
mation; but the first offers it by way of information
about men, and the second by way of a description of
the universe. Information may be given about a pre
dicate by the use of a different kind of copula; as in
" no lack of hospitality is found among Baltimoreans."

An inclusion is changed into the equivalent exclusion
by changing the sign of the predicate. When an exclusion
is to be made into an inclusion, it is a matter of indiffer
ence which of its terms is regarded as predicate ; every
exclusion contains within itself two inclusions, of which
each is the converse by contraposition of the other.
That is to say,

AV B = A^<B = B^<^..

With this copula, therefore, the consideration of the con
version of propositions is rendered unnecessary. So also
is the consideration of the quantification of the predicate.
With the copula -< the subject and predicate have un
like quantity, or, more exactly, the quantity of the
subject is universal and that of the predicate is indeter
minate ; -< means either equal to or less than. But
with the copula V the quantity of both subject and pre
dicate is universal, and with its denial V both subject
and predicate are taken in part only.

The copula -< must be taken in an inverted sense
according as subject and predicate are taken in exten
sion or in intension ; but the copula V possesses the same
meaning, whatever interpretation one gives to the terms
which it separates. The proposition men are animals
means that all the individuals, man, are contained among

28 ON THE ALGEBRA OF LOGIC.

the individuals, animal ; but that the qualities which dis
tinguish an animal are contaiued among the qualities
which distinguish a man. The proposition no stones are
plants means that the objects denoted and the qualities
connoted by the term stone are inconsistent with the
objects denoted and the qualities connoted by the term
plant. It is to be remembered that every term is at
once a sum of objects and a product of qualities. If
the term a denotes the objects al9 «2> az . . . and con
notes the qualities al9 «2, «3 . . . , then

a = al a2 + . . .
and the full content of the proposition no a is b is

But the full content of the proposition all a is b can be
expressed only by the two statements

a1 + a2+... + ai-< bL + b2+ bs+ .. . and ftft. . .&-< a^ag...

where the i objects a are identical each with some one of
the objects b and they qualities ft are identical each with
some one of the qualities a.

If p denotes a premise and c a conclusion drawn from
it, then

p V G (m)

states that the premise and the denial of the conclusion
cannot go together ; and

p V o (n)

states that the premise is sometimes accompanied by the
falsity of the conclusion. It is hardly necessary to men
tion that (w) is satisfied by either the truth of the con
clusion or the falsity of the premise, and that (n)
implies that both the premise and the negative of the
conclusion must, at some time, be true.

CXN" THE ALGEBRA OF LOGIC.

The word inference (or consequence) implies proceed
ing in a definite direction in an argument, — either from
the premise to the conclusion, or from the negative of the
conclusion to the negative of the premise. The argu
ment p v c may be called an inconsistency. It is a
form of argument into which the idea of succession does
not enter ; it simply denies the possible co-existence of
two propositions. An inconsistency between two propo
sitions is equivalent to each of two equivalent conse
quences, and a consistency to each of two equivalent
inconsequences ; or

c=p-<c =

The copulas V and V with the symbol oo give means for
expressing the total non-existence and the partial exist
ence of expressions of any degree of complexity. Pro
positions with the symbol 0 do not occur in this algebra.

(16') x v °° = "x does
not, under any circumstances,
exist."

A universal proposition does
not imply the existence of its
subject; therefore x \7 0 = "x
(if there is any x) is not
non-existent," — a proposition
which is true whatever x may
be.

(16°) x V oo = "x is at
least sometimes existent."

A particular proposition
does imply the existence of its
subject ; therefore x V 0 = " x
exists, and at the same time
does not exist," — a propo
sition which is false whatever
x may be.

Since the symbol 0 will not appear at all in proposi
tions expressed with these copulas, it will not be neces
sary to write the symbol co . I shall therefore express
" there is no x " simply by x V-

ON THE ALGEBRA OF LOGIC.

(170 a\Jb = aby

abc \/=a\/bc = ca\/b = ...

To say that no a is b is the
same thing as to say that the
combination ab does not ex
ist.

(17°) aVb
abc V = aV bc

To say that some a is b is
the same thing as to say that
the combination ab does at
least sometimes exist.

The factors of a combination which is excluded or not
excluded may be written in any order, and the copula
may be inserted at any point, or it may be written at
either end. The proposition abc y de may be read " abc
is-not de," " cd is-not abe" " abe is-not do, — that is, is
either not d or not c," etc. Any 0, 1, 2, 3, 4, or 5 of the
letters may be made the subject and the others the predi
cate, and the positive or the negative universal copula
may be used ; or there are in all 2.32, = 64, different
ways of putting the above proposition into words.

If a is a proposition, a \j states that the proposition is
not true in the universe of discourse. For several pro
positions, abc v means that they are not all at the same
time true ; and the way in which they are stated to be
not all at the same time true depends on the character
of the universe. If it be the universe of the logically
possible, then pl p2 c \/ states that pi and p% may be
taken as the premises and c the conclusion of a valid
syllogism. It is the single expression in this system for
a proposition which in the system of inclusions appears
in the several forms

from the premises the conclusion follows ; if the conclu
sion is false, one at least of the premises is false ; from
one premise may be inferred either the conclusion or the
contradictory of the other premise, and from one pre
mise and the contradictory of the conclusion follows the

ON THE ALGEBRA OF LOGIC. 31

contradictory of the other premise. If the universe
which is understood is the universe of what is possible
in accordance with the laws of nature, then ab V denotes
that the simultaneous truth of a and b is a contradiction
of those laws. That x and y stand in the relation of
cause and effect may be expressed by xy \j. If a? is a
certain position and y its attendant acceleration, the
above proposition states that the position and the ab
sence of the acceleration are not found together; that
from the position may be inferred the acceleration, and
from the absence of the acceleration may be inferred the
absence of the position. If a V I means that Greeks are
brave, and c \j d means that the megatherium is not
extinct, then

(a\/l)\/(c\j d)

affirms that the co-existence of these two propositions is
excluded from the universe of what is actually true. In
like manner, according to the character of the universe
of discourse, a V 6 denotes either that the two proposi
tions are logically consistent, or that they are possibly
co-existent, or that they have actually been at some mo
ment of time both true.1

ALGEBRA OF THE COPULA.
By the definition (1), we have

(18) (a = b) = (ayt)(ayb).
Since also

(a = 5) = (*v!)(av»),

it follows that

(19) (a = b) = (a = I).
In particular,

(20) (ab = 0) = (ab = oo) = (ab y oo) ;

1 The thorough-going extension of the idea of a limited universe to the
relations between propositions is due to Mr. Peirce.

32 ON THE ALGEBRA OF LOGIC.

for the exclusions to which each equation is equivalent
are (ab V oo) (db V 0),

and ab V 0 is a proposition of no content.

The principles of contradiction and excluded middle
are therefore completely expressed by

(70 aa v- 1 (7°) a+a V.

In any symbolic logic there are three subjects for con
sideration, — the uniting and separating of propositions;
the insertion or omission of terms, or immediate infer
ence ; and elimination with the least possible loss of
content, or syllogism.

On uniting and separating Propositions. — From the
definitions of logical sum and logical product applied
to terms and to propositions we have the following iden
tities : —

(210 (a

for the first member of the
equation states that a does
not exist and that b does not
exist; and the second mem
ber states that neither a nor

(21°)

for the first member of the
equation states that either a
exists or b exists; and the
second member states that
either a or b exists.

b exists.

In both cases, a and b may be logical expressions of
any degree of complexity. A combination of any num
ber of universal propositions, or an alternation of any
number of particular propositions, is then expressed as a
single proposition by taking the sum of the elements of
the separate propositions. This is the only form of in
ference (if it should be called inference at all) in which
the conclusion is identical with the premises. The equa
tions (21') and (21°) are not in reality two distinct

ON THE ALGEBKA OF LOGIC.

equations ; they are, by (19), one and the same equa
tion; since, by (13), the negative of («v) (^ V) is
(a V) + (ft V), and the negative of a + by is a+ ft V.
They are each equivalent to the two inconsistencies,

(a v) (b v) V 0 + ft V)
C J

(» v) + 0 v) v 0 + ^ v)-

There is no single expression in this algebra for a sum
of universal propositions or a product of particular pro
positions.

To express that the propo
sitions, some a is b and some c
is d, are not both at the same
time true (or that it is true
throughout the universe of dis
course that either no a is b or

To express that neither of
the propositions, no a is b and
no c is d, is true (or that it is
true throughout the universe
of discourse that both some a
is b and some c is d), we must
write

else no c is d), we must write

(a y ft) (c V d) y>
And the expression for the corresponding particular
propositions which follow from these universals is

(a \7 ft) + (c v d) v ;
that is, there is some time

when either no a is b or else
no c is d.

(a V b) (c yd) V ;

that is, there is some time
when both some a is b and
some c is d.

On inserting and dropping Terms. — The following in
consistencies are immediate consequences of the defini
tions of the sum and the product : —

(23) (abc V) v (^ V)-

The first asserts that the total non-existence of several
things is inconsistent with the existence of some of
them ; the second asserts that the total non-existence

ON THE ALGEBRA OF LOGIC.

of something, as ab, is inconsistent with the existence of
some part of it, as ab which is c. They are not two dis
tinct inconsistencies, however ; either may be derived as
a consequence from the other. These inconsistencies,
when put into the form of inferences, become —

(22°) If a + b V,

then a + b + c V ;

(23°) If abc V,
then ab V.

That is to say, given a par
tial inclusion, factors may be
dropped and parts of a -sum
may be introduced, but not
without loss of content.

(22') If a + b + c V?
then a + b y >

(23') If ab V>

then abc y .

That is to say, given a uni
versal exclusion, factors may
be introduced and parts of a
sum may be dropped, but not
without loss of content.

As a particular case of both of the inconsistencies
(22) and (23) we have

(a yb)(cyd)y(ac\fb + d).1 I.

If into the expression which is affirmed not to exist,
ab + cd, we introduce the factor c + a ; and if from the
product, acb + acd + ab + cd, we drop the parts of a sum,
ab + cd, — there remains ac (6 + eT), the existence of
which is inconsistent with the non-existence of ab and
cd. Since there is no difference between subject and

predicate,

(ayb)(c~yd)y(a + c\f bd)

is an inconsistency of quite the same nature as I. For
the expression of /. in words we have —

Ia. It is not possible that what is common to several
classes should have any quality which is excluded from

1 In its affirmative form, "if a is b and c is d, then ac is bd," this is
Theorem I. of Mr. Peirce's paper on the Algebra of Logic (XXL). As
pointed out by Mr. Venn, it was first given by Leibnitz : " Specimen de-
monstrandi," Erdmann, p. 99.

ON THE ALGEBRA OF LOGIC. 35

one of them. If, for example, no bankers are poor and
no lawyers are honest, it is impossible that lawyers who
are bankers should be either poor or honest.

In this way the theorem is put into words in terms of
a quality which is excluded from a class. It is a pro
perty of the negative copula that it lends itself equally
well to the expression of propositions wholly in exten
sion and wholly in intension, and also with the subject
taken intensively and the predicate extensively. We
should have in words, in these cases respectively —

Ib. If several classes are respectively excluded from
several others, no part of what is common to them can
be included in any of the others ;

Ic. If several qualities are inconsistent respectively
with several others, their combination is not consistent
with any of the others.

Id. It is not possible that a combination of several
qualities should be found in any classes from each of
which some one of those qualities is absent. If, for
example, culture is never found in business men nor
respectability among artists, then it is impossible that
cultured respectability should be found among either
business men or artists.

The inconsistency I. is the most general form of that
mode of reasoning in which a conclusion is drawn from
two premises, by throwing away part of the information
which they convey and uniting in one proposition that
part which it is desired to retain. It will be shown that
it includes syllogism as a particular case. The essential
character of the syllogism is that it effects the elimina
tion of a middle term, and in this argument there is no
middle term to be eliminated.

When combinations of any number of terms are given
as excluded, a proposition with which they are inconsis-

36 ON THE ALGEBRA OF LOGIC.

tent can be formed by taking any number of terms out
of each and uniting them as a sum and denying their
co-existence with the product of the terms which re
main. If _ _
abc v? plh V?

affirm that no American bankers are uncharitable and
that no Philadelphia lawyers are dishonest, then it is im
possible that any Philadelphia bankers are either un
charitable Americans or dishonest lawyers; that any
uncharitable and dishonest lawyers are either Philadel-
phians or American bankers ; that any bankers who are
also Philadelphia lawyers are either uncharitable Ameri
cans or dishonest, etc. Any, none, one, two, or three,
terms from the first premise may be taken to form the
sum with any, none, one, two, or three, terms from the
second premise; there are, therefore, sixteen different
conclusions to be drawn in this way from these two
premises, — of which dbcpTil \/ is the least, since it has
dropped the most information, and abc + plh y is the
greatest, since it has dropped none of the information.

The inconsistency I. may be put into an inference in
four different ways, according as both universals, one
universal, one universal and the particular, or the parti
cular alone, is taken as premise and the negative of what
remains as conclusion. There are, therefore (when I.
contains the smallest possible number of propositions),
four distinct forms of inference, or progressive argu
ment, with no middle term, in each of which the con
clusion is a diminished conclusion. The factors of I.
are, in general, one particular and any number of uni
versals. The number of distinct forms of progressive
argument which can be made out of an inconsistency
between n propositions of which n — 1 are universal, by
taking 1, 2, . . or n — 1 of the universal propositions with

ON THE ALGEBRA OF LOGIC.

or without the particular proposition as premise and the
negative of what remains as conclusion, is 2(^ — 1).
Argument by way of inconsistencies, therefore, what
ever may be thought of its naturalness, is at least
2 (n — 1) times more condensed than argument in the
usual form.

When I. is made into an inference in such a way that
one conclusion is drawn from two premises, we have,

if the premises are both uni
versal,

(24') « V *

c \7 d
.'. ac \/ b rh d

If no bankers have souls
and no poets have bodies, then
no banker-poets have either
souls or bodies.

if the premises are one uni
versal and one particular,

(24°) ayb

ac v # + d
.'. c v d

If no Africans are brave
and some African chiefs are
either brave or deceitful, then
some chiefs are deceitful.

On Elimination. — In (24') there is no elimination,
and in (24°) there is elimination of the whole of the
first premise and part of the second. The most common
object in reasoning is to eliminate a single term at a time,
— namely, one which occurs in both premises. Each of
these inferences gives rise to a form of argument, as a
special case, by which that object is accomplished,—
the premises being on the one hand both universal, and on
the other hand one universal and the other particular.
The inconsistency I. becomes, when d is equal to 5, and
hence b + d equal to oo,

(a v V) (c V &) (ac v <*>) V>
or

(a v 1) (^ v c) (G V a) V- -^

Given any two of these propositions, the third proposi
tion, with which it is inconsistent, is free from the term

ON THE ALGEBRA OF LOGIC.

common to the two given propositions ; a, &, and c are,
of course, expressions of any degree of complexity.
The propositions ma ^/x + y,xy~\jc + n, for instance,
arc inconsistent with ma V c + n; any number of terms
may be eliminated at once by combining them in such a
way that they shall make up a complete universe.

When any two of the inconsistent propositions in II.
are taken as premises, the negative of the remaining one
is the conclusion. There are, therefore, two distinct
forms of inference with elimination of a middle term,
special cases of (24') and (24°). If we write x for the
middle term, we have

(25') a\/ x

b\/x
.-. ab v-

The premises are
a (b + 1) x \/
(a + a) bx v ;
and together they affirm that

ab (x + x) + abx + abx v>

ab + abx + a£x V-

Dropping the information con
cerning x, there remains

ab y.

The information given by the
conclusion is thus exactly one

half of the information given
by the premises (Jevons).

(25°) a v x

bV x
.-. ba V.
The second premise is

bx (ax+ax) V,
which becomes, since there is

no ax,

bx (a + x) V,
bxa\f.

Dropping the information con
cerning x, there remains

baV.

This conclusion is equivalent
to

ba V x + #;

but the premises permit the
conclusion

ba V z;

hence the amount of informa
tion retained is exactly one half of the (particular) infor
mation given by the premises.

ON THE ALGEBRA OF LOGIC.

Elimination is therefore merely a particular case of
dropping irrelevant information.

When a and b are single
terms, (25') is the doubly uni
versal syllogism, and it is the
single form in which that
syllogism appears in this alge
bra. When it is translated
into syllogism with an affirm
ative copula, it is necessary
to consider the four variations
of figure which are produced
according as x or x is made
subject or predicate. The
eight moods in each figure
correspond to the eight varia
tions of sign which may be
given to a, b. and x. All the
rules for the validity of the
doubly universal syllogism are
contained in these : —

(1) The middle term must
have unlike signs in the two
premises.

(2) The other terms have
the same sign in the conclu
sion as in the premises.

When a and b are single
terms, (25°) is the universal-
particular syllogism, and it
is the only form of that syllo
gism in this algebra. It can
be translated into eight differ
ent forms of syllogism with
unsymmetrical copula, accord
ing as x is made subject or
predicate of either premise,
and according as a or b is
made subject of the conclu
sion. The eight moods of the
major and minor particular
syllogism in each figure corre
spond to the eight variations
of sign which may be given
to a, b, and x. All the rules
for the validity of the uni
versal-particular syllogism are
contained in these : —

(1) The middle term must
have the same sign in both
premises.

(2) The other term of the
universal premise only has its
sign changed in the conclu
sion.

Those syllogisms in which a particular conclusion is
drawn from two universal premises become illogical
when the universal proposition is taken as not implying
the existence of its terms.1

1 McColl : Symbolical Reasoning, — Mind, no. xvii. Peirce : Algebra
of Logic, — Am. Journal of Math., vol. iii.

40 ON THE ALGEBRA OF LOGIC.

The argument of inconsistency,

(a v 1) (J> V <0 0 V «) v> II-

is therefore the single form to which all the ninety-six
valid syllogisms (both universal and particular) may be
reduced. It is an affirmation of inconsistency between
three propositions in three terms, — such that one of the
propositions is particular, and the other two are univer
sal ; and such that the term common to the two universal
propositions appears with unlike signs, and the other two
terms appear with like signs. Any given syllogism is
immediately reduced to this form by taking the contra
dictory of the conclusion, and by seeing that universal
propositions are expressed with a negative copula and
particular propositions with an affirmative copula. Thus
the syllogisms Baroko and Bokardo,1

All P is M, Some M is not P,

Some S is not M, All M is £,

.-. Some S is not P, /. Some S is not P,

are equivalent respectively to the inconsistencies

(P V M) (S V M) (S V P) V,
(MM P) (MV S) (S V P) V.

1 If there were ever any occasion to use the mnemonic verses of syllo
gism, it might be worth while to put them into a form in which each word
should bear the mark of its figure, as well as of its mood and its method
of reduction. By some slight changes in the words, the first, second,
third, and fourth figures might be indicated by the letters r, t, I, and n
respectively : —

(r) Barbara, Cegare, Darn, Ferioque prioris.

(t) Cesate, Camestes, Festive, Batoko secundee.

(I) Tertia, Dalipi, Disalmis, Dalisi, Felapo.

(I) Bokalo, Feliso, habet ; quarta insuper addit,

(n) Bamanip, Camenes, Dimanis, Fesanpo, Fesison.

ON THE ALGEBRA OF LOGIC. 41

It is then possible to give a perfectly general rule, easy
to remember and easy of application, for testing the
validity of any syllogism, universal or particular, which
is given in words. It is this : —

Rule of Syllogism. — Take the contradictory of the con
clusion, and see that universal propositions are expressed
with a negative copula and particular propositions with
an affirmative copula. If two of the propositions are
universal and the other particular, and if that term only
which is common to the two universal propositions has
unlike signs, then, and only then, the syllogism is valid.

For instance, the syllogism —

Only Greeks are brave,
All Spartans are Greeks,
Therefore all Spartans are brave,

is equivalent to the inconsistency —

Non-Greeks are-not brave,
Spartans are-not non-Greeks,
Some Spartans are not-brave,

which fails to stand the test of validity in two respects, —
the term brave appears with unlike signs and the term
Greeks with like signs. The syllogism -

All men are mortal,
Some mortals are happy,
Therefore some men are happy,

is equivalent to the inconsistency —

Men are-not immortal,
Some mortals are happy,
Men are-not happy,

and it is not valid for the same reasons as before, — the

42 ON THE ALGEBRA OF LOGIC.

term mortal appears with unlike signs, and the term
men with like signs.

When #, 5, and x are expressions of any degree of
complexity, (25') and (25°) still furnish the only means
for the elimination of x. For instance, if

(ab -f cd) x v

and

(a + c)x + bfy,
then

abc + dcd + bf y,

is all that can be said without reference to x. And if

(ab + cd)x + bfy
and

(a + c)x y}

then the conclusion, irrespective of #, is

(ab + cd) a + c + bf y,
or

ac b

If the premises consist of propositions about proposi
tions, then any proposition which it is desired to drop
may be eliminated in accordance with these two rules.

Syllogisms are the inferences, with elimination, which
are obtained by taking two of the propositions of I. as
premises and the other as conclusion. When one propo
sition only is taken as premise, the conclusion is an
alternation of propositions ; and, as a special case, a
single arbitrary term (instead of two or none) may be
introduced. We have —

ON THE ALGEBKA OF LOGIC.

ayb,

(26')

or, in words, if no a is b, then
either no ac is either b or d,
or else some c is d. If no
Africans are brave, then either
some chiefs are deceitful, or
else no African chiefs are
either brave or deceitful.
When c = x, d — l, this be
comes
(27') _«v*

.•. (a v x) + (5 V x)-
If no Africans are brave, then
either no Africans are Chinese
or else some Chinese are not
brave.

(26°)

acVb + d',

or, in words, if some ac is
either b or d, then either some
a is b or some c is d. If some
African chiefs are either brave
or deceitful, then either some
Africans are brave or some
chiefs are deceitful. When
b = d = x, this becomes

(27°) ac V

.-. (a v a;) + (c V ac).

If some lawyers are bankers,
then either some lawyers are
honest or some bankers are
dishonest.

Inference from Universal to Particulars. — Dimin
ished statement and that particular form of diminished
statement which is syllogism are the only reasoning pro
cesses that are valid when a universe which contains
nothing is included among possible universes, — that is,
when it is taken as possible that both x and x may be at
the same time non-existent. When that universe is ex
cluded, — when the postulate "z and non-z cannot both
be non-existent" is taken as true, — one other form of
reasoning is possible. That postulate is expressed by

(x v) V (x V)>

which is equivalent to the two inferences, " if x does not
exist, then non-z does exist," and " if non-z does not
exist, then x does exist ; " or, from the total non-exist
ence of any expression whatever may be inferred the
existence of some part at least of its negative. If

44 ON THE ALGEBRA OF LOGIC.

a(b+c)v, then « + 5c V , and iid + le y , then a (6 + <?) V ;
or,

If # is a proposition, & V 5, then non-# is its denial,
a\/b', and the postulate states that a proposition cannot
be both true and false at the same time.
From the proposition

ab v
follows, in this way,

ab v > that is, a + 5 V.

The complete convention in regard to the existence of
terms is therefore : the particular proposition a V b im
plies the existence of both a and b ; the universal propo
sition a y b does not imply the existence of either a or 5,
but it does imply the existence of either a or b. The
necessity of the convention (if it should be called a con
vention) is even more evident when a and b are proposi
tions ; in that case it is equivalent to saying that two
propositions cannot be true together unless each is at
some time true, and that they cannot be not true to
gether unless one or the other is at some time false.

Mr. McColl has pointed out that from u all a is 5,"
" some a is b " does not follow, because there may not
be any a. But from

aB v
it does follow that

a5V ; that is, ab + ab + ab V ;

or from " all a is b " it does follow that one at least of
the propositions " some a is 6," " some not-a is 5,"
" some not-a is not 5," is true. From any universal prop
osition follows some one at least of the three particular
propositions which it does not contradict. If a is known

ON THE ALGEBRA OF LOGIC. 45

to exist, then " some a is b" follows from " all a is b "

by a syllogism : —

aB v

aaV
.'. ab V

From " no sea-serpents have gills " we cannot infer that
there are some sea-serpents which are without gills, un
less it is known that there are some sea-serpents ; but
we can infer that either there are some sea-serpents
without gills, or there are some things, with or without
gills, which are not sea-serpents, or else there is nothing
in the universe.

EESOLUTION OF PROBLEMS.

Rule. — Express universal propositions with the nega
tive copula and particular propositions with the affirma
tive copula, remembering that a — b is equivalent to

ab + ab Y,

and that its contradictory, a is not equal to 6, is equiva
lent to

al + ab V.

From a combination of universal propositions, the con
clusion, irrespective of any term or set of terms to be
eliminated, x, consists of the universal exclusion of the
product of the coefficient of x by that of the negative
of x, added to the excluded combinations which are free
from x as given. If the premises include an alternation
of particular propositions, the conclusion consists of the
partial inclusion of the total coefficient of x in the par
ticular propositions by the negative of that of x in the
universal propositions, added to the included combina
tions which are free from x as given.

46 ON THE ALGEBRA OF LOGIC.

If there is any reason for expressing a universal
conclusion with an affirmative copula or a particular
conclusion with a negative copula, it can be done by
taking any term or set of terms as subject and the
negative of what remains as predicate.

The premises may also contain an alternation of any
number of universal propositions. If either

(p v x) or (q v x) or (r y z),
and if at the same time

am y x,
then

am (p + q + rz) V

is the conclusion irrespective of x. When a combina
tion of particular propositions is included among the
premises, the conclusion consists of a combination of
the same number of particular propositions. From

(pyx) fev^)
(a Vx) (bV x),

may be inferred the two propositions,

(ay fq) (by pq).

From particular propositions by themselves no con
clusion follows, otherwise than by simply dropping un
necessary information.

Particular premises may be attached to the universal
premises by the conjunction or instead of the conjunction
and. In that case no elimination is possible (except
what can be done between the universal propositions by
themselves), and a conclusion can be obtained only by
means of "the postulate, P. If either (ayb and c \j cT)
or (jg v h and i vy), then the conclusions are gh + ab y,
ij + ab V, gli + cd V, ij + cd V. In general, then, the
premises may consist of a combination or an alter-

ON THE ALGEBRA OF LOGIC. 47

nation of universal propositions (two cases), or of par
ticular propositions (two cases), or a combination or
an alternation of universal propositions united as a sum
or a product to a combination or an alternation of
particular propositions (eight cases).

It is apparent that logical notation would be improved
by the addition of another sign, by means of which an
alternation of universal and a combination of particular
propositions might be expressed as a single propo
sition, — a sign such that

(p + x) sign qy sign rz \/

should mean that some one of the expressions p + x, qy,
rz, is totally non-existent, and its contradictory,

(p + x) sign qy sign rz V,

should mean that all of these are, at least in part,
existent.

The plan of treating a set of universal premises as a
command to exclude certain combinations of the terms
which enter them is due to Boole ; no adequate exten
sion of his method so as to take in particular propo
sitions is possible, without the use of some device which
shall be equivalent to a particular copula. Boole's
method of elimination between universal propositions
is to put x first equal to 0 and then to 1 in the given
function, and to take the product of the results so ob
tained. The only difference between this rule and that
which I have given (which is Prof. Schroder's) is that
it first introduces x into those terms which are already
free from it, and then proceeds to eliminate it from all.
The value of the function

ax _|_ ix + c, or ax + bx + c (x + x),
for x — 0 (in this case b + c) is the coefficient of x, and

48 ON THE ALGEBRA OF LOGIC.

its value for x — 1 (in this case a + c) is the coefficient
of x. I have shown that the method is not an invention
of modern times, but that it is nothing more than a rule
for working the syllogism,

All b is x, No a is x, .-. No a is.ft,

when a, 5, and x are not restricted to being simple
terms. With the unsymmetrical copula, there are four
different forms of pairs of universal propositions which
make possible the elimination of x (XXI., p. 39), and
for its elimination between a universal and a particular
proposition it would be necessary to consider eight
different forms, corresponding in all to the twelve dis
tinct forms of syllogism.

If the result which remains after elimination is of the
form

am + bm + c y (c)

(where m is the term in regard to which information is
sought, and where all the letters are expressions of
any degree of complexity), and if there is any reason for
being dissatisfied with the conclusion as it stands, —
" no m is a, no b is not m, and there is no c," — m may
be made subject and predicate respectively of two affirm
ative propositions, " all b is w, and all m is a." If it
be desired to express the conclusion without any repe
tition, then we must first state what is true without
regard to m, — in this case,

ab + c \7>

"there is no ab nor <?," — and then this information
must be used to diminish the propositions in m. The

identities

a = a (ab + c + ab + c)

b = b (ab + c + ab + c)
become, when there is no ab + <?,

(8')

ON THE ALGEBKA OF LOGIC. 49

a = a . ab + c = #5c,
b = b . ab + c = bac ;
and hence, instead of

a v m>t by m,
it is sufficient to write

ale v m>) boo y m ;
or, affirmatively,

All m is b + c + a,
All bac is m.

Prof. Schroder expresses in terms of m such a con

clusion as

am + &?H + c (m + m) = 0,

by means of the formula

[0 + c) m + (b + c) m = 0]
= [m = all (b + c) + some a + c] [&& + c = 0].

The first factor of the second member of the equation is
equivalent to the propositions,

All m is b + c + ac,
All (6 + c) is w,

Some a + c is m ;

that is, it contains the propositions of the first member
(the first diminished by ab + c = 0 and the second not),
but it contains in addition the particular proposition
" some a + c is wi," which is a legitimate inference
from " no (a + c) is m" only if a + c is known to exist.
A more condensed equational form of the conclusion
am + bm + c\/ is

(m = all bac + some TJac) (ab + c = 0).
Boole reaches the same conclusion, ((7), but he does

50 ON THE ALGEBKA OF LOGIC.

it by an extremely circuitous route. Nothing could well
be simpler of application or more evident than this
rule of Prof. Schroder's, and there is no reason why
one should not place implicit confidence in it, in an
algebra in which particular propositions are not taken
as implying the existence of their terms. It contains
the solution of what Mr. Jevons calls the " inverse log
ical problem," and which he solves by a process " which
is always tentative, and consists in inventing laws and
trying whether their results agree with those before us "
(XXII., p. 252). It makes all reference to tables and
machines quite superfluous. It seems to have been
overlooked by the latest expositor of Boole's system, —
Mr. Venn. He says that Boole's method of getting his
conclusion is " a terribly long process ; a sort of ma
chine meant to be looked at and explained, rather than
to be put in use ; " and that if ever we do feel occasion
to solve such a problem, it can be done most readily
" by exercise, so to say, of our own observation and
sagacity, instead of taking, and trusting to, a precise
rule for the purpose of effecting it " (XXIII., p. 316).

But Boole's form for the conclusion (besides being
not quite legitimate in this algebra) is not that which
is most natural or most frequently useful. It is, more
over, suited only to a logic of extension, and it would
be difficult to interpret intensively. The very simple
device which may be substituted for ,it is to make use
of the same method for getting back from excluded
combinations to affirmative propositions which was em
ployed in passing from the given affirmative propositions
to the excluded combinations : if

All b is m = b\/ m,
then

b m = all 1) is m.

ON THE ALGEBRA OF LOGIC. 51

In this way the conclusions are given in the form which
has been adopted by Mr. McColl. Complicated prob
lems are solved with far more ease by Mr. McColl than
by Mr. Jevons ; but that is not because the method of
excluded combinations is not, when properly treated,
the easiest method. A method of implications, such
as that of Mr. McColl, is without doubt more natural
than the other when universal premises are given in
the affirmative form, but the distinction which it pre
serves between subject and predicate introduces a rather
greater degree of complexity into the rules for working
it. An advantage of writing abc y instead of dbc = 0
is that the copula can be inserted at any point in the
excluded combination, and that elimination can be per
formed on the premises as they are given, when they
have been expressed negatively, without first trans
posing all the members to one side. Without some
thing corresponding to a contradictory copula, particular
propositions cannot be treated adequately, and compli
cated propositions of either kind cannot be simply
denied. With it, the contradictory of " all a is all £,"
that is, " it is not true that all a is all &," is al + db V ;
that is, " either some a is not b or some b is not a."
And the contradictory of

abc + abc + abc y
is

abc + abc + abc V ;

that is, some one at least of the given combinations is
in existence.

EXAMPLES.

1. (By Mr. Venn in Mind for October, 1876.) The
members of a board were all of them either bondhold
ers or share-holders, but no member was bond-holder

52 ON THE ALGEBRA OF LOGIC.

and share-holder at once ; and the bond-holders, as it
happened, were all on the board. What is the relation
between bond-holders and share-holders ?

Put

a = member of board,
b — bond-holder,
c = share-holder.

The premises are evidently

a v be + %Cj
bya;

and taking the product of the coefficient of a by that of
a, we have

b (be + 5c) Y>
or

bey.

The required relation is, therefore,

No bond-holders are share-holders.

2. (XXII., p. 283.) What are the precise points of
agreement and difference between two disputants, one
of whom asserts that (1) space (a) = three-way spread
(5), with points as elements (<?) (Henrici) ; while his
opponent holds that (2) space = three-way spread, and
at the same time (3) space has points as elements ?

(a = be) = (aB + ac + abc \/), (1)

ac y. (3)

They both assert that

a5 + ac -f- «#c v?

and the second asserts in addition that

dbc Y j

ON THE ALGEBRA OF LOGIC. 53

that is, that a three-way spread which had not points as
elements would be space.

3. (XVI., vol. x. p. 21.) From the premises

bxyc (cl + i/)e
ab v x (3, + e) c

a + b +

deduce a proposition containing neither x nor y.

The term y does not occur at all ; hence y can be
eliminated only by dropping the parts which contain it.
There remain

acct + alj (<l + e) y x,

bcde v x ;
and taking the product of the first members we have

abcde \/.

4. (XXIIL, p. 310.)

Given ^ ~ > , find xz in terms of a and c.
yz = c ) '

The equations are equivalent to the exclusions

xya -\-xa-\-ya v?
yzc -\-yc + zc\/;
and after elimination of y there remains

xa + zc + %ac + z<w V- O9)

Collecting the predicates of xz and xz, we have

xz v dc + «c,

(?)
ic + * V ac'

54 ON THE ALGEBEA OF LOGIC.

Prof. Schroder's formula, (7, p. 49,

If m v x and m y y> then m = all y + some x,
gives, in this case,

xz — all ac + some (ac + ac)

= all ac + some do.
If it were required to find xz + xz, we should have

xz -\-xz\/ ac,
xz + xz v ac + ac ;
whence

cci + ieg = all (ac + ac) + some (ac + ca + ac)

= all (ac + ac) + some ac.
It is evident that (jp) cannot be inferred from (g).

5. (Educational Times, Feb. 1, 1881, 6616. By W. B.
Grove, B. A.) The members of a scientific society are
divided into three sections, which are denoted by a. b, c.
Every member must join one, at least, of these sections,
subject to the following conditions : (1) Any one who
is a member of a but not of 5, of b but not of c, or of c
but not of a, may deliver a lecture to the members
if he has paid his subscription, but otherwise not;
(2) one who is a member of a but not of c, of c but not
of a, or of b but not of a, may exhibit an experiment
to the members if he has paid his subscription, but
otherwise not; but (3) every member must either
deliver a lecture or perform an experiment annually
before the other members. Find the least addition to
these rules which will compel every member to pay his
subscription or forfeit his membership, and explain the
result.

ON THE ALGEBRA OF LOGIC. 55

Put x = he must deliver a lecture, y = he must per
form an experiment, and z = he has paid his subscrip
tion. Then the premises are

ale v (a)

al + Ic + cd v xz (1)

ac + cd + db y yz (2)

xyy. (3)

It is required that z be excluded from all that part of
the universe from which it has not already been ex
cluded ; namely, from the negative of

(al + be + cd) x + (ac + ca + ab) y + ale + xy,
which is, by the second rule for getting the negative,

(ale + abc + x) (ale + ac + y) (a + b + c) (x + ?/),
or

abcx + ac^y.

Hence the desired " least addition to the rules " is
abcx -f- acxy y z,

or, " No one who has not paid his subscription can be
a member of all three sections and deliver a lecture,
or of a and c and perform an experiment without lec
turing."

6. (III., p. 237. Proposed for simpler solution by
Mr. Grove, Educational Times, April 1, 1881.) A num
ber of pieces of cloth striped with different colors were
submitted to inspection, and the two following observa
tions were made upon them : —

(a) Every piece striped with white (w~) and green (#)
was also striped with black (5) and yellow (?/), and
vice versa.

56 ON THE ALGEBRA OF LOGIC.

(5) Every piece striped with red (d) and orange (r)
was also striped with blue (w) and yellow, and vice
versa.

It is required to eliminate yellow, and to express the
conclusion in terms of green.

The premises are

W9 — fy> dr = uy ;

and by (18') they are equivalent to the exclusions

dr (u + y) + uydr y.

Collecting the coefficients of y and y we have
bwg + udr y y,
Wff + dryy;

and taking the product of the left-hand members we
have

uwgdr + bdr (w + g) y,

which is to be added to that part of the premises which
does not contain y ; that is, to

wig + dru y.
Concerning g we have

g y w (5 + udr), bdr y g ;

or, with the affirmative copula, by (30),

g -< w + bu + bdr, bdr -< g.

The first is equivalent to Boole's conclusion when that
is reduced by dru = 0. For the second Boole gives only
bdrwu -< g.

To solve this problem by Mr. Jevons's method, it
would be necessary to write out the one hundred and
twenty-eight possible combinations of seven terms, and
to examine them all in connection with each of the

ON THE ALGEBRA OF LOGIC. 57

premises. As Mr. Jevons himself says : " It is hardly
possible to apply this process to problems of more than
six terms, owing to the large number of combinations
which would require examination" (XIII., p. 96).

7. (III., p. 146). From the premises

xz (v + wy + wy) \f

v xw (yz + yz) \J
x(v + y) (zw + zw) V
(x + vy) (zw + zw) V

it is required, first, to eliminate v ; second, to express
the conclusion in terms of x ; third, in terms of y ;
fourth, to eliminate xm, fifth, to eliminate y.
The terms which involve v are

xz + xw (yz + yz) + y (zw + zw) y v, x (zw + zw) y v ;

whence, taking the product of the left-hand members,
we have only

xzyw v, (a)

which is to be added to that part of the premises which
does not contain v, — namely, to

xz (wy + wy) + xy (zw + zw) + x (zw + zw) \/.
Collecting the parts which contain x and x we have

x \7 zw + yzw, (b)

x \7 zw + zw + zwy. (c)

The negative of the second member of (<?) is, by (14),
zw + zwy^ hence, by (18'), these two exclusions are
equivalent to the identity

x = zw + zw + zwy) (^)

x = zw + yzw.

58 ON THE ALGEBRA OF LOGIC.

No part of the conclusion lias been dropped in .(ft)
and (c) ; hence the propositions which concern y may be
taken from them. They are

y xzw, xzw V $

y -< x + z + w, xzw

These exclusions yield nothing upon the elimination
of y ; hence the only relation between a?, s, and w is,
from (5) and (c),

cezw; + Jezw + 5J£«# y". (y)

These conclusions are the same as those of Mr. McColl,
and they are equivalent to those of Boole and Schroder.
Prof. Wundt (XVIII., p. 356) accidentally omits (a) in
getting the conclusions in regard to y, and they are in
consequence altogether wrong. He remarks that Schro
der has treated the problem in a partly coincident
manner. I do not find that Mr. Jevons has treated it
at all.

8. Six children, #, b, c, d, e, /, are required to obey
the following rules: (1) on Monday and Tuesday no
four can go out together ; (2) on Thursday, Friday,
and Saturday, no three can stay in together; (3) on
Tuesday, Wednesday, and Saturday, if b and c are to
gether, then a, b, e, and / must remain together ; (4)
on Monday and Saturday b cannot go out unless either
d, or c, e, and / stay at home, b and/ are first to decide
what they will do, and c makes his decision before «,
d, or e. Find («) when c must go out, (/3) when he
must stay in, and (7) when he may do as he pleases.

Let a be the statement that a goes out, and a the
statement that he stays in, etc. Then we have for the
first two premises

ON THE ALGEBRA OF LOGIC. 59'

M+ T\J abed + bcde + . . . (1)

Th + F+ S v ale + aid + . . . (2)

The third premise excludes from certain days the com
bination in which b and c are both out or both in, ex
cept when a, 6, e, and/ are together ; that is,

T+W+ S~y(bc + bc) abef+ abej
V (T>c + bc) (a + 5 +

or, finally,

T+W+ S\/tca + bce + bcf+ bca + bee + bcf. (3)

The last premise is, for Monday,

M~ybd(c + e+f). (4)

On Saturday, c, e, and / cannot all stay at home, by
(2) ; therefore, this part of the premise is

Sybd. (4')

The first thing required is the elimination of a, d, and
e. That part of the premises which is already free from
those letters is

(3')

Nothing can be eliminated between (1) and (2), because
MTh = 0, etc.

For the same reason, d cannot be eliminated between
(4) and (2) ; and therefore the premise (4) must bo
simply dropped, a and e can be eliminated at once by
combining (3) with (1) and with (2). From (3) and
(1), we have respectively

(T + W+ S)bcya + e,

60 ON THE ALGEBEA OF LOGIC.

and taking the product of the right-hand members and
the sum of the left-hand members, we have

T(4o)v. (5)

From (3) and (2) we have respectively

(T+W+ S)Zc ya + e,

whence, in the same way,

S(lo) v. (6)

By combining (4') with that part of (2) which does not
contain a, e, or 25, and does contain cl, — namely, with

we obtain

Sybfc. (7)

The conclusion required is then contained in (2'), (3'),
(5), (6), and (7). But the information given in regard
to S and T may be somewhat simplified by collecting
their predicates. We have

S\/lcf+ lcf+ Ic + bcf+ be/,
or

Sytc + bf, (8)

and

Sr/5 (9)

which with

Th + fytcf, (2")

Wybcf+lcf, (3")

form the entire conclusion. Collecting the subjects of
c and c} we have

(Th + F) lf+ (T+ W) lf+ Stye (a)

Tb + Wbfy c (b)

ON THE ALGEBRA OF LOGIC. 61

where the last proposition is already independent of c,
and where c cannot be eliminated between (#) and (6).
The conclusion may be expressed in words in this way :
(a), if on Thursday or Friday b and / are both at
home, or if on Tuesday or Wednesday / goes out with
out 5, or if b stays at home on Saturday, then c must
go out; (/3), if b goes out on Tuesday, or if b goes out
without / on Wednesday, then c must stay at home ;
(7), whether c goes out or stays in, b does not go out
without /on Saturday.

OX THE CONSTITUTION" OF THE UNIVERSE.

The number of combinations in the complete develop
ment of n terms is 2n. In any actual universe of things,
any one of these combinations may be either present or
absent ; hence the number of different ways in which a
universe may be made up out of n things is 22W. The
following Table gives the sixteen possible constitutions
of the universe with respect to two terms. The sign 1
indicates the presence of the combination at the head
of which it stands, 0 its absence. With the aid of
the dual notation, applied to logical algebra by Mr.
Franklin,1 each case may be defined by a number ; it is
only necessary to attribute powers of two as weights to
the different combinations, and to describe each arrange
ment by the sum of the weights of the combinations
which are present in it. If we take the, combinations of
a and b in the order a&, db, al,dl, then 4, or 0100, de
notes that the combination aB is present, and nothing
else ; 9, or 1001, that dl and ab are present and al and
db are absent, etc.

1 Johns Hopkins University Circular, April, 1881.

ON THE ALGEBRA OF LOGIC.

al
8

aH
4

ab
2

If a is animal and b is black, then the 5th case is that
of a universe made up of black animals and animals
which are not black ; in the 12th case the things which
are wanting are black animals and black things which
are not animals, — that is, there are no black things in
this universe ; the 15th case is the actual universe with
respect to the terms animal and black ; the 0-case is a
universe in which nothing exists. If the material uni-

ON THE ALGEBRA OF LOGIC. 63

verse is the subject of discourse, and if a means matter
and I means indestructible, then the existing state of
things is described by 4 ; indestructible matter exists,
and what is not indestructible matter docs not exist.
This Table is given by Jevons (XIII., p. 135) ; but he
does not take account of non-existent terms, and hence
all but seven of the sixteen cases (all but 6, 7, 9, 11,
13, 14, 15) are considered by him to be logical absurdi
ties. If a and b are propositions, then case 9 is a
universe in which they are true together and false to
gether, and in which the time during which a is true
is identical with the time during which b is true, either
logically or extra-logically. The 0-case is a universe in
which no proposition is true. Two cases the sum of
whose characteristic numbers is 15, — as 5 and 10, or
0101 and 1010, — have been called by Prof. Clifford
complementary cases : what exists in one is what does
not exist in the other.

To exactly define the constitution of any universe, it
is necessary to state, in regard to each combination, that
it is present or that it is absent. The simple laws which
every two terms obey are therefore four in number,
being partly universal propositions and partly particu
lar ; except in the 0-case, where all the universal propo
sitions are true, and in case 15, where all the particular
propositions are true. The perfectly symmetrical uni
verses are thus the universe in which there is nothing
and that in which there is some of everything. For
case 8, we have

(a \/b)(a\fb)(ayP)(ay £),
and for case 13

(a y b) (a v b) (a y 5) (u V I).
When two simple or compound statements cannot be

THE ALGEBRA OF LOGIC.

converted into each other by any interchange between
the terms which enter them (including negatives of
terms), they are said to belong to different types. The
universal propositions in two terms are of six different
types. None, one, two, three, or four of them may be
true, and it is only in the case where two are true that a
difference of type is produced by the way in which the
propositions are selected. Those two may be taken so
that one letter has the same sign in both or not. Thus
we may have either,

ab + ab y,
that is,

or
that is,

ab + ab \/,

a — .

The following Table gives the six types, the proposi
tions which define them, and the universes which belong
to each type : —

Type.

Universal.

Particular.

Cases.

(a V £) (a V b) (a V 5) (a V 5)

II.

a\/b

(a V V) (a V I) (a V I)

8, 4, 2, 1

III.

a v

(a Vb)(aV I)

12, 3, 10, 5

IV.

a = b

(a Vb)(aV I)

6, 9

a-\-b\/

a\fl

7,11,13,14

VI.

$,_{_<2_L-£»-L-7J\7

I. and VI. are complementary types ; and so are II. and
V. The universes complementary to III. and IV. are

ON THE ALGEBRA OF LOGIC. 65

of types III. and IV. respectively. Six is the number
of types of a universe in two terms, when all the par
ticular propositions which the universal propositions do
not deny are known to be true. If one takes account of
combinations of alternations and alternations of com
binations of both particular and universal propositions,
the number of types is largely increased.

A race of beings which always completely defined its
universe would have the above four-fold statements for
its forms of expression. The eight propositions which
are used by the race which exists are not complete
definitions of a universe, but they are symmetrical;
each has an eight-fold degree of ambiguity. "No a is 5"
denies the existence of the combination a5, but it leaves
it doubtful whether, of the remaining combinations, none,
any one, any two, or all three exist. " Some a is 5,"
which affirms the existence of the combination ab, re
stricts the universe to some one of the eight cases, — 1,
3, 5, 7, 9, 11, 13, 15. If, however, propositions are
taken in the other sense, — if positive (affirmative)
propositions are taken as implying the existence of the
subject and negative not, — then they do not include all
possible states of things with symmetry. The negative
universal and the positive particular propositions cover
eight cases each, as before ; but of the positive universal
a v I takes in the four cases 1, 3, 9, 11, and a \j b the
six cases 1, 4, 5, 9, 12, 13 only, and their contradictories,
the negative particular, have respectively a twelve-fold
and a ten-fold degree of ambiguity.

On the other hand, a race of beings which had the
greatest possible variety of expression would be able to
speak with any degree of ambiguity at pleasure. It
would have a distinct propositional form for restricting
the universe to any one, one of any two, one of any

66 ON THE ALGEBEA OF LOGIC.

three, etc., of the possible cases ; or its entire number
of propositions in two terms would be 216 or 216 — 1,
according as one counts or does not count the case in
which nothing is said. All the 65,536 or 65,535 things
which can be said without using any other terms than
theologians and scientists, for instance, the existing
race is able to say, without very much difficulty, by
combinations and alternations of its Aristotelian and
Morganic propositions. To say that either no scientists
are theologians (0, 2, 4, 6, 8, 10, 12, 14), or some theo
logians are not scientists (3, 7, 11, 15), or some of those
who are not theologians are scientists and some are not
scientists (13), or else everybody is a theologian (1), is
to make a statement of fourteen-fold ambiguity, — to
limit the constitution of the universe under considera
tion to some case exclusive of 5 and 9. The contradic
tory of a statement of the form

(a v 6) + (a v &) + (» V 5) (« V^H @ v)
is, by (13),

(a V b) (a v b) (a v 5 + a y 5) (5 V) ;

and to affirm that there are some theologians who are
scientists, and that there are no theologians who are
not scientists, and that either all scientists or else all
non-scientists are theologians, and that not everybody
is a theologian, is to affirm that either 5 or 9 furnishes
the complete description of the universe with respect to
the terms scientist and theologian.

In three terms the number of combinations is 23, the
number of possible universes is 223, = 256, and the num
ber of possible propositions with all degrees of ambiguity
is 2256. The types of universal propositions have been
given by Mr. Jevons (XIII., p. 140), but the number is
increased when single terms as well as combinations

ON THE ALGEBRA OF LOGIC.

are permitted to be non-existent. Prof. Clifford's
method for obtaining types (" Essays and Lectures. —
On the Types of Compound Statement involving Four
Classes") is not difficult when applied to these terms.
It takes account of terms which do not exist, and the
number of types which he gives for four terms, 396,
would be different on any other hypothesis. The prob
lem would certainly be extremely difficult if such state
ments as Mr. Jevons calls contradictory were excluded.
Prof. Clifford's solution takes account of combinations
only of universal propositions. The number of types
of alternations only, and of alternations and combina
tions of particular propositions only, is also 396, and
the entire number is in this way raised to 4,396 ; but
the determination of the number for mixed universal
and particular propositions and for mixed alternations
and combinations of them is still in the region of un
solved problems.

In three terms, the number of types of combinations
of universal propositions is twenty-six, — six four-fold,
eight less than four-fold, and eight more than four-fold.
The types of more than four-fold statement may be
obtained by taking those combinations which are not
excluded by the types of less than four-fold statement.

LESS THAN FOUR-FOLD.

I.
II.

III.

IV.
V.

VI.

VII.

VIII.

abc

abc + abc

abc + alJc + abc

abc + abc + aBc

abc + abc + abc

ON THE ALGEBBA OF LOGIC.
FOUR-FOLD.

IX.

XI.

XII.

XIII.

XIY.

abc + abc + abc + abc
abc + abc + abc + abc
abc + abc + abc + abc
abc + abc + aBc + abc
abc + «5c + abc + «5c
o&c + abc + a&c + a5c

When condensed, these exclusions appear in the fol
lowing form. The Arabic numbers give the correspond
ing types in Mr. Jevons's Table.

. .

XXII.

II.

abc

XXI.

. .

a + b + c

III.

XX.

. .

a + b

IV.

(ab + al) c

XIX.

. .

ab + ab + c

abc + abc

XVIII.

ab + be + ca

VI.

(a + b)c

XVII.

. .

ab + c

VII.

ab + abc

XVI.

ab + ab + abc

VIII.

abc + (ab + ab)c

XV.

(a + b)c+(al + ab)c

IX.

ab + be + ca

. .

XI.

ac-\- be

XII.

Obb + ab

XIII.
XIV.

14
6

a (be + Ic) +a (lc + be)
abc + (a + 5) c

The exclusions

IV., XVIII., XI., XII., XIII.,
are equivalent respectively to the identities

XIV.

ab = c.

ON THE ALGEBRA OF LOGIC. 69

In these Tables, the letters may represent propositions
as well as terms ; of the 256 ways in which three propo
sitions may be put together they give the 22 which are
of distinct type. Case V., for instance, is the case in
which three propositions, pl9 p^ p^ are affirmed to be not
all at the same time true and not all at the same time
false ; or, in other words, it is known that some one of
them is true and some one of them is false. In case
XVIL, pl and p2 are not true together, and pz is not
true at all. When the universe under discussion is the
logical universe, the Tables serve to enumerate the 22
possible types of argument between three propositions.
In case IX., ply p^ ps are propositions so related that
from the truth of any one the falsity of the other two
can be inferred ; in case XI., they are such that if two
of them are both false or both true, the third is there
fore false ; and, conversely, if that is false, the others
are therefore either botli true or both false. The syllo-
gism pip2pz v> ig °^ tne tyPe H- The argument "if
either some animals covered with fur are black or some
black things not covered with fur are animals, then
some animals are black," — that is,

(abx V) + (abx v) \/(ab y),

which is of the form (j?i +^2)p3 v> — belongs to type
VI. ; and the identity,

(a \7 b) (c V d) — (ab +ca> \7)>

belongs to type XIV. In order to find actual arguments
of all the 22 types, it would probably be necessary to go
into some hyper-universe where the laws of thought are
different from those under which we reason.

70 OX THE ALGEBRA OF LOGIC.

BIBLIOGRAPHY.

I. George Boole : The Mathematical Analysis of Logic, being an
essay towards the Calculus of Deductive Reasoning. 1847.

II. Boole : The Calculus of Logic. Cambridge and Dublin Math.
Jour., Vol. III., 1848.

III. Boole : An Investigation of the Laws of Thought, on which
are founded the Mathematical Theories of Logic and Probabilities.
1854.

IV. Hermann Grassmann : Lehrbuch der Arithmetik. Berlin,
1861.

V. "W. S. Jevons : Pure Logic, or the Logic of Quality apart from
Quantity. London and New York, 1864.

VI. C. S. Peirce : On an Improvement in Boole's Calculus of
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VII. W. S. Jevons : The Substitution of Similars, the true Princi
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VIII. C. S. Peirce : Description of a Notation for the Logic of
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IX. Robert Grassmann : Die Formenlehre oder Mathematik.
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X. A. J. Ellis : On the Algebraical Analogues of Logical Rela
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XI. Arthur Cayley : Note on the Calculus of Logic. Quart. Jour,
of Math., Vol. XL

XII. Ernst Schroder : Lehrbuch der Arithmetik und Algebra fur
Lehrer und Studirende. I. Bd. : Die sieben algebraischen Opera-
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XIII. W. S. Jevons : Principles of Science. London, 1874. Third
edition, 1879.

XIV. Ernst Schroder : Der Operationskreis des Logikkalkuls.
Leipzig, 1877.

XV. J. Delboeuf : Logique Algorithmique. Liege et Bruxelles,
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XVI. Hugh McColl : The Calculus of Equivalent Statements, and
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Vol. X., 1878, Vol. XL, 1880.

XVII. Gottlob Frege : Begriffsschrift, eine der arithmetischen
nachgebildete Formelsprache des reinen Denkens. Halle, 1879. Re-

ON THE ALGEBKA OF LOGIC. 71

viewed by Ernst Schroder in Zeitschrift fur MathematiJc und Physik,
1880.

XVIII. Wilhelm Wundt : Logik, eine Untersuchung der Princi-
pien der Erkenntniss und der Methoden wissenschaftlicher Forschung.
I. Bd. : Erkenntnisslehre. Stuttgart, 1880.

XIX. J. Venn : On the Diagrammatic and Mechanical Repre-
sentation of Propositions and Reasoning. Phil. Mag., July, 1880.

XX. J. Venn : Symbolic Reasoning. Mind, July, 1880.

XXI. C. S. Peirce : On the Algebra of Logic. Am. Jour, of Math.,
Vol. III., September, 1880.

XXII. W. S. Jevons : Studies in Deductive Logic. London, 1880.

XXIII. John Venn : Symbolic Logic. London, 1881.

NOTE. — In the foregoing article "combination" has been used as
synonymous with "multiplication." In the following article, "combi
nation " is used as including both multiplication and addition.

ON A NEW ALGEBRA OF LOGIC.

BY 0. H. MITCHELL.

THE algebra of logic which I wish to propose may be
briefly characterized as follows : All propositions — cate
gorical, hypothetical, or disjunctive — are expressed as
logical polynomials, and the rule of inference from a set
of premises is : Take the logical product of the premises
and erase the terms to be eliminated. No set of terms
can be eliminated whose erasure would destroy an ag-
gregant term. So far as the ordinary universal prem
ises are concerned, the method will be seen to be simply
the negative of Boole's method as modified by Schroder.
The reason is, that the terms which the propositions
involve are virtually all on the right-hand side of the
copula, instead of all on the left-hand side, as in Boole's
method.

Attention is especially called to the treatment here
given of particular propositions (of which there is intro
duced a variety of new kinds) which is homogeneous
with that of universals, the process of elimination being
precisely the same in each case. For the sake of clear
ness it may be well to state at the outset that I use
addition in the modified Boolian sense, — that is, x + y
= all that is either x or y.

ON A NEW ALGEBRA OF LOGIC. 73

§ 1. Logic has principally to do with the relations of
objects of thought. A proposition is a statement of such
a relation. The objects of thought, among which rela
tions may be conceived to exist, include not only class
terms but also propositions. The statement of a rela
tion among propositions is a proposition about proposi
tions, which Boole called a secondary proposition. But
every proposition in its ultimate analysis expresses a
relation among class terms. The universe of class terms,
implied by every proposition or set of propositions, may
be limited or unlimited. Two class terms, a, £, are
defined as the negatives of each other by the equations

a + b = U,
ab = 0,

where U is the symbol for the universe 'of class terms.
Two prepositional terms, a, /3, are defined as the nega
tives or contradictories of each other by the equations

a. + $ = GO,

«p=o,

where oo is the symbol for the universe of relation, or
for "the possible state of things." Mr. Peirce uses oo
indifferently as a symbol for the universe of class terms,
or for the universe of relation, but in the method of this
paper it seems most convenient to have separate sym
bols. We can speak of " all of " or " some of " £7, but
hardly, it seems to me, of " all of " or " some of " the
universe of relation ; that is, the state of things. For
this reason oo seems an especially appropriate symbol
for the universe of relation.

The relation implied by a proposition may be con
ceived as concerning " all of " or " some of " the uni
verse of class terms. In the first case the proposition

74 ON A NEW ALGEBRA OF LOGIC.

is called universal ; in the second, particular. The rela
tion may be conceived as permanent or as temporary ;
that is, as lasting during the whole of a given quantity of
time, limited or unlimited, — the Universe of Time, — or
as lasting for only a (definite or indefinite) portion of it.
A proposition may then be said to be universal or par
ticular in time. The universe of relation is thus two-
dimensional, so to speak ; that is, a relation exists among
the objects in the universe of class terms during the
universe of time.

The ordinary propositions neglect the element of time ;
and these will first occupy our attention.

Let F be any logical polynomial involving class
terms and their negatives, that is, any sum of products
(aggregants) of such terms. Then the following are
respectively the forms of the universal and the particular
propositions : —

All Vis Fj here denoted by F19
Some CT is.*; « « Fu.

These two forms are so related that

= oo

that is, Fl and Fu are negatives of each other ; that is,
\) = Fu. The two propositions Fl and J\ satisfy the

one equation

and are " contraries " of each other. Whence, by taking
the negative of both sides, we get

FU + FU= oo;

that is, Fu and Fu are " sub-contraries " of each other.
The line over the F in the above does not indicate the
negative of the proposition, only the negative of the

ON A NEW ALGEBRA OF LOGIC. 75

predicate, F. The negative of the proposition Fl is not
jFi, but (^), which, according to the above, = Fu.

The Aristotelian propositions are represented in this
notation as follows : —

(a + 5)x = All of U is « + 5 = No a is 5, . . . . .#.

(a#)M = Some of U is ab = Some a is b, .... /.

(a + b\ = All of £7 is a + £ = All a is 5, .... A.

(ab)u = Some of Z7 is a5 = Some a is not ft, . . 0.

By substituting a, I for a, 5 throughout we get the four
complementary propositions of De Morgan. If these
two forms be applied to the sixteen possible sums of ab,
aS, db, aS, there results the following

TABLE OF PROPOSITIONS.

(ab + al + ab + ab)^ .......... (0)

(at + ab + «5)1 ............ (ab)

(ab + al + ab), ............ (a5)

(ab + ab + ab)1 ............ (ab)

(ab + al + ab\ ............ (al)

(ab + afyi ............ («5 + ab)

(ab + ab^ ............ (a5 + al)

(ab + al\ ............ yn + afy

(aB + ab\ ............ (al + a^)

(a5 + al\ ............ (ab + ab)

............ (ab + ab + a^)u

(al\ ............ (ab + ab + aJ).

............ (a5 + ab + 05).

............ (ab + a5 + ab)u

ON A NEW ALGEBRA OF LOGIC.

Opposite propositions are negatives of each other.
The Table reduced to its simplest form becomes

REDUCED TABLE.

(U),

(0)«

(a + 5V

(aft

(a + b],

. lab)

(a + 5s),

. (ab}u

(a + b),

. (ab)u

(a),

(a\

(b}

(b)

. . . (aB + ab)

(ob + db)l . . .

. . . (ab + ab\

(b)

(a\

(a}

(ab},

.(a + %)

(06}

(a + b)

(ab\

. (a 4. J)

(«5)i -

. (a + b),

(0)i

• (U}u

If three terms be treated in a similar way we get
2.223,= 512, different propositions. With n terms the
total number is 2.22".

The propositions (0)i and (0)tt assert that there is no
universe of discourse, and are false in every argument,
since a universe of class terms greater than zero is to be
pre-supposed. Their negatives (£7)M, (^)i are there
fore true in every argument. The eight propositions of
De Morgan occur in lines 2, 3, 4, 5 of the Table.

ON A NEW ALGEBRA OF LOGIC. 77

Since the universe of class terms is supposed greater
than zero, the dictum de omni gives

-*i-<^;
that is, " all Z7is J7" implies " some U is I7."

To say " no UisF " is evidently the same as to say " all

U is F;" that is, F0 = F19 and since a proposition whose
suffix is 0 is thus expressible in a form with the suffix
equal to 1, each suffix used will be supposed greater than
zero. The suffix u in Fu is taken to be a fraction or part
of U less than the whole ; that is, " some of " U. In the
proposition "some U is jP" it is not denied that all

U may be J7, but the assertion is made of only a part
of U. Thus u is taken as greater than zero and less
than 1, or U. When u is written as a suffix of different
propositions in the same argument, it is not meant that
the same part of U is concerned in each case. F± is writ
ten for convenience instead of Fv. Sometimes Fe will
be written as a form inclusive of both the forms F: and
Fu ; that is, e will be considered as having either of the
two values 1 or u.

For inference ~by combination of such propositions we
have the following simple rules : —

The conclusion from the
product of two premises is the
product of the predicates of
the premises affected by a suf
fix equal to the product (in
extension) of the suffices of
the premises. Thus

The conclusion from the
sum of two premises is the
sum of the predicates of the
premises affected by a suffix
equal to the sum (in intension)
of the suffices of the premi
ses. Thus

* This is Mr. Peirce's sign for the copula of inclusion, being an abbrevia
tion of ^. It is read "is," "is included under," or "implies." The
following formulae are sometimes made use of in this paper : (a -< b)
= (2> -< a) = (06 = 0) = ( oo = a -\- b), where a> = the universe of dis
course. Also, (a -< b) (x -< y) -< (ax -< by).

78 ON A NEW ALGEBRA OF LOGIC.

When both premises are ( umver^al 1 the relation be-

l particular 3

tween the j Product 1 and the conclusion is equality ;

I sum
otherwise, the relation is -<, an implication. Thus

(1) F& = (FG)

(2) F,Gu-<

(3) FUGU -< oo.

FU+GU= (F+G)U, (10
Fu+G,-< (F+G)U, (20
F1+G1-<(F+G\.(3')

These formula? are so evident as hardly to need explana
tion. (1) means

(U=F) (U=G) = (V-=

and it follows from the definition of logical multiplica
tion. By taking the negative of both sides, and chang
ing F, & to F, G-, we get (!')• The law of the suffices
in (!') is u 4- u = u, or some + some = some. (2) means

(U=F) (u=G) = (u =

and follows also from multiplication. The law of suf
fices is 1 u = u ; that is, Uu = u. Since G-1 -< (7«, (20
follows from (!'). The law of the suffices is u + 1 = w ;
that is, " all of " or " some of " = " some of," which is ad
dition in an intensive sense. In formula (3) there can
be no inference when nothing is known about the rela
tion of the two suffices ; that is, Fu Gu —<^ oo. If it be
known that u and u' have any common part, then for
this common part FUG-U> -< (FG)UU,. Thus if u = f Z7,
and u'=%U, then Fu G-u, -< (FQ-\,, , where u" = uu'=^U.
Since we evidently have (FG-)U^^ FUGU, we get by
contraposition the formula (3'), which means in words
" ' either all U= F, or all U=Gr' implies ' all U— either

Having regard to (1) and (I'), it will be seen that

ON A NEW ALGEBRA OF LOGIC. 79

the most general proposition under the given conditions
is of the form

where F and G are any logical polynomials of class
terms, II denotes a product, and S denotes a sum.

If F and G be any of the sixteen polynomials involv
ing two class terms a, 5, it is interesting to notice that
any proposition, Z^IIG-J, can be reduced to the sum
of products of the eight propositions of De Morgan.
Thus, referring to the Table on page 76, any proposition
Fl in the first column is equivalent (1) to the product
of one or more of the propositions 2,3, 4, 5, — that is, E,
A, E\ A' (the two universal propositions added by De
Morgan to the classic two being represented by Efy A') ;
and any proposition Gu of the second column is equiva
lent (I/) to the sum of one or more of the propositions
7, (9, J', 0', the two accented letters representing the
particular propositions added by De Morgan. Thus
Fl = U«, and II Gu = 112 '/3 = 2U/3, where a is one of
the four universals of De Morgan, and /3 is one of the
four particulars. Thus

Thus, for example, the proposition

(a + b\ (ab + a5)j -f- («5)1 + (&)u (# + ^)i>
when reduced, becomes

{(«&)« + («5)M + («&)„} (a + &)1 (a + 5)T

4- (« + 5)! (a + 5)1(a + &)1+ {(«£)„+ (a5)u}(a

In like manner it may be shown that if F, G, etc. be
logical functions of any number of class terms, a, 5, c,
etc., the general proposition

80 ON A NEW ALGEBRA OF LOGIC.

may be reduced to a function of the eight propositions
of De Morgan of the form

where p, etc. are the eight propositions.

Propositions united by + form disjunctive propo
sitions. A hypothetical proposition, " if a, then /3," or
a -< £, where a and j3 are themselves propositions, is
evidently equivalent to the purely disjunctive propo
sition a + ft. Thus " if a is be, then cd is e " means

(a + &<Oi-< 0 + 5 + e)j;
which is the same as

(aB + ae)u + (e+S + e)19

which may be put into words in one way as follows:
" some a is either non-6 or non-c, or all d which is non-c
is e." The preceding formulas are examples of inference,
by combination of propositions ; that is, of inference from
a product or from a sum of propositions.

Inference by elimination will now be considered. It
will only be necessary to consider the fundamental form
jPe, where e may be either 1 or u. If I7 be a polynomial
of the class terms, a,b,c,... x, y, z, then x, y, z may be
eliminated from Fe by erasure, provided no aggregant
term is thereby destroyed. That is,

Ft -< F,',

where F1 is what remains of F after the erasure. Thus
(ax + bcxy + dcz + db)e -< (a + be + dc + db)e .

The reason is obvious. To say that " (all or some) U
is dx, or bcxy, or etc.," is saying by an obvious implica
tion that " (all or some) U is a, or be, or etc." Fe
means (all or some) U -< F, and the erasure of a fac
tor of a monomial term of F simply increases the extent

ON A NEW ALGEBRA OF LOGIC. 81

of the term ; therefore the predicate F is not diminished,
and (all or some) U -<^ F', — that is, FJ is a valid in
ference. F1 is really the sum of the coefficients of x, y,
z in F, and is obviously a factor of F. The other factor
of F is F + P ; for F' (F + F') = F, and F + F' is seen
to contain no factor independent of x, #, 2, since on
erasing x, ?/, 2, the result is .F' + F', = U. If one of the
aggregant terms of F contain no letters except those to
be eliminated, then its coefficient is U, and Ft' will in
this case be a nugatory result. Thus from (a + led),
b, c, d, be, Id, or cd can be eliminated, but not a, ab, ac,
ad, ale, aid, aed, or led. As already stated, this alge
bra is the negative of Boole's as modified by Schroder,
so far as universal premises are concerned. Thus Boole
multiplied propositions by addition, and eliminated by
multiplying coefficients. The method here employed
multiplies propositions by multiplication, and eliminates
by adding coefficients. When many eliminations are
demanded in a problem, the advantage in point of
brevity of this method over Boole's is of course greatly
increased.

Before considering some illustrative examples, another
kind of inference is to be noticed ; namely, inference by
predication ; that is, the finding what a given proposition
says about a given term, simple or complex. The rule
is : Multiply F by the given term, m, or add m to F. The
resulting coefficient of m in mF, or the residue of F after
adding iii and reducing, will be the predicate of m. Thus

Fe -< (m = mF).,
or Fe~< (m + F)t.

The first means, "if U=F for all or some U, then
m = mF for all or some U" and the result is obviously
obtained by multiplying both sides of U= Fby m. The

82 Otf A NEW ALGEBRA OF LOGIC.

second relation means, "if U = F for all or some U,
then U = m + F for all or some Z7," and the result is
obtained by adding m to both sides, remembering that
U + m = U. We have, of course,

(m + F)e = (m + mF)e = (m = mF)e.

I now give the solution of the well-known problem of
Boole, " Laws of Thought," p. 146. The premises are,
remembering that (« = &) = (« + b^ (a + 5)^

(x + z -f vyw + vwy)lf

(v + x + w + yz + yz)19

(x + vy + w£ + wz\(xy + va; + w« + wz)^

Multiplying the premises together, and dropping v from

the result, we get

(wxz + wxz -f- wfl32 + iZ?xy + wxyz)19 = say J^.
The four results asked for by the problem are

(1) (x + wz + ws +

(2) (wS + wz + ^ + w;y +

(3) (y + ^^^ + wxz + wxz

(4) (wa? + i/5S + xz + i

The first gives the predicate of x in terms of #, ^, w,
being the same as x -< wi + w^ + wjy, and is obtained by
adding x to ^ and reducing. The second is the relation
among y, 25, w, and is obtained by dropping x from F
and reducing. The result (U\ shows that no relation
is implied among ^, 2, «0 alone. The third gives the
predicate of y in terms of #, 2, w, and is obtained by
adding # to .F and reducing. The fourth is the relation
implied among x, z, w, and is obtained by dropping y
from F and reducing. The relation (3) is not in its
simplest form, since the implied relation (4) among #, 2, w

ON A NEW ALGEBRA OF LOGIC. 83

has not yet been taken into account. Since (p. 81)
we have F = F' (F + F1) , where F] is what remains
of F after erasing ?/, and F + F' is that factor of F
which contains no factor independent of ?/, we get
Fl = F^F-i- F'\. The first factor FJ is (4), and from
the second factor we get (y + F+F'\ as the simplest
form of (3), that is,

\ U i ^\ c ' * / 1

Ordinary syllogism appears in this method as follows :
The mood Barbara becomes

b being eliminated by dropping it. The moods Darii,
Datisi, Disamis, and Dimaris are all

(ab)u (5+ c)x -< (abc)u-< (ac\.
The premises of the mood Darapti are

(m +p\(m + s)lf = (m + sp\\

but there is no conclusion independent of the middle
term m, since m cannot be eliminated. In inferring the
conclusion I from these two premises logicians have
virtually included a third premise (w)tt, that is, " some
of U is m," or " there is some ;«." This with the pro
duct of the other two gives " some 8 is p ; " that is,

(m + sp)l(m)u -< (spm)u -< (sp)u.

In the same way, the premises of Felapton and Fesapo
are

(m +p)1(m + 5)x = (m + sp)i>

and m cannot be eliminated here. With the additional
premise (m)u we get " some * is not p ; " that is,

(m + sp\(m)u -< (spm)u -< (sp)H.
The premises of Bramantip are

(p + m)^ + s)1 = (sp + sm + mp\ -< (s + p\\

84 ON A NEW ALGEBKA OF LOGIC.

that is, the conclusion is not " some s is p" but " all p is
«," or " all 5 is p" the proposition A1. Here, again, the
conclusion " some s is p " has been reached only by the
virtual inclusion of a third premise, " there is some p"
that is, (p)u. Then we have

(sp + sm + mp)i(p)u ~< (smp)u ~< (sp)u>

This is the same thing as to say that a particular con-
elusion cannot be drawn from universal premises, since
a particular proposition implies the existence of its sub
ject, while a universal does not.1

As another illustration of the method, I solve the
problem in Boole's "Laws of thought," p. 207. The
premises are

(w + xyz + xyz + xyz\,

Their product is

[f{wy + w (p~jr + pqr + pqr) + xyz + zyz + xyz (pTjr +pqr
+ wx {pqyz + pqys + pqryz}\, say F1 ,

which contains everything implied in the premises. The
results asked for are

1. (r + t + z)l9 whence t -< r + z

2. (r+t + y)19 " t-<r + y

3. (U\9

1 Mr. Peirce and others.

ON A NEW ALGEBRA OF LOGIC.

(t + x)19
(p + <i +

6. (t + yz + yzr)}

7. (t -J- yz + y%)\ >

whence
tt

whence

t -< x

y-<p + 2

rt -< yz
tz -< yr
ry -< t

t-< U
,z + yz-< t.

The relations in the first column are each obtained by
dropping from Fl the letters not concerned in the qurasi-
tum. Each predicate in the second column is obtained
by multiplying its opposite X" by its subject. The re
sult 4 disagrees with that obtained by Boole.

The two examples taken from Boole have dealt ex
clusively with universal propositions. The following is
of a different kind : —

What may be inferred independent of x and j from
the two premises, " either some a that is x is not y, or all
d is both x and y; " and "either some y is both b and x,
or all x is either not y or c and not b " ?

The premises are

(axy)u + (d, + xy\,
(bxy)u+ (x + y + le\.
By multiplication we get

(axy)u (bxy\ + (bxy\ + (axy)u + (fix + ay + Icoi + lcxy)l .
Whence, dropping x, y and reducing, we get

which may be interpreted in words, " there is some 5, or
a, or else all d is c and not 6."

86 ON A NEW ALGEBRA OF LOGIC.

From this result we may further eliminate c. Elimi
nating c, we get

(b + a)u+ (3 + l)19

which means " either b or a exists, or no d is 5."

The analogy bettveen class and propositioned terms. —
Hitherto in the consideration of Fl and Fu the polynomial
F has been regarded as a function of class terms a, £,
etc. Suppose a, b, etc. to be prepositional terms like
Fl and Fu, and call the resulting polynomial no longer F,
but $. Then the suffices of ^ and £M cannot be in
terpreted any longer as referring to the universe of class
terms, since the prepositional terms F19 7^, etc., of which
$ is a function, are supposed to have already suffices
with this meaning. The suffices of $x and $M can only
be interpreted then as referring to the universe of the
time during which the complex or secondary proposition
# is supposed to be true. Then, if F denote the uni
verse of time,

<£>! means " $, during all F," or " all V -< <£,"
$v " "$, " some F," or "some F-< <3>."
In otlier words

^ means " $ is always true,"
<£» " " ^ is sometimes true,"
where " always " refers to the universe of time, V,

Owing to the similarity between class terms and prepo
sitional terms with respect to the operations of multipli
cation and addition, it follows that all that has been
said in regard to inference from propositions like F^ Fu
holds equally with ^ and $v . Thus

i »

ON A NEW ALGEBRA OF LOGIC. 87

So in regard to elimination, any set of terms can be
eliminated by neglect, provided no aggregant term is
thereby destroyed.

§ 2. Propositions of Two Dimensions.

Let U stand for the universe of class terms, as before,
and let V represent the universe of time. Let I7 be a
polynomial function of class terms, a, 6, etc. Then let
us consider the following system of six propositions :

Fuv, meaning "some part of U, during some part of V, is F"
Ful) " " some part of U, during every part of V, is F"
Flv, " "every part of U, during some part of V, is F?
Fufl, " " the same part of U, during every part of F, is F"
Fiv't " "every part of U, during the same part of V, is F"
Fn, " "every part of U, during every part of F, is F."

By thus introducing the element of time, three varieties
of the proposition Fu are distinguished, — Fuv, Ful, FU>1.
Thus in speaking of the people of a certain village during
a certain summer (Z7= village, V= summer), "some
of the Browns were at the sea-shore during the sum
mer" may mean either that some of them were there
during a part of the summer, or that some of them were
there during every part of the summer, — not necessarily
the same persons, — or that the same persons werepthere
during the whole summer. These three meanings are
here denoted respectively by (&«)«»? (£«)«i> (&s)«i- Three
varieties of F1 are also distinguished, — Fn, Flv, Flv,.
Thus " all the Browns were ill during the year " may
mean either that every one was ill during every part of
the year, or that every one was ill during some part of
the year, — not necessarily the same part, — or that every

ON A NEW ALGEBKA OF LOGIC.

one was ill during the same part of the year. These
three meanings are denoted respectively by (I + i]
P + ^P + fV

The dictum de omni gives the following relations
among these six propositions : —

ul,FlvFulFuv) and F^ +
and since same is included under some, we have

Flv,-< Fln and F^ -< Ful.
The following pairs of propositions,

Fuv and Fu ,Ful and F* , FU>1 and Fl9,
satisfy the two equations

and the members of each pair are therefore the negatives
or contradictories of each other. Thus if F= I +\ it is
seen that (U\v and (5 + i)u are contradictories ; that is,
" either some of the Browns were not ill during some
part of the year, or they were all ill during every part
of the year," and both cannot be true. An example
of the second pair is (li)ul and (5 + ?)ll/; that is, " either
some of the Browns were ill during every part of the
year (not necessarily the same persons during the whole
year) or at some particular time none of them were
ill," a*nd both cannot be true. An example of the third
pair is (fo')ttl and (b + i)lv, " either the same Browns
were ill during the whole year, or it was true for each

* The natural first thought is that FU, Fui, FIV, Fuv form a system of
propositions by themselves, but it is seen that FW and Fu>i must be added
to the system, in order to contradict Fu} and Flv. Mr. Peirce pointed out
to me that these propositions are really triple relatives, and are therefore
six in number. Fn, for instance, means "F is a description of U during
V" See the Johns Hopkins University Circular, August, 1882, p. 204.

ON A NEW ALGEBRA OF LOGIC. 89

part of the village during some part of the year that none
of the Browns were ill," and both cannot be true.

Since from A -< B we get A + B = GO and AB = 0,
so from Fn -<^ Fw

we get Fvv + Fuv = GO,

and ^ii^ii = 0;

hence Fn and "Fn are " contraries " of each other, and
Fuv , Fuv are " sub-contraries." In the same way F& -< Flv

gives Fvl + Flv = GO,

and Flv,Fv>! = 0 ;

that is, Fw and F^ are contraries, and Fvl, Flv are sub-
contraries. The line over F affects only F, not the
suffices. Thus the negative of Fn would be written
(3y, not Fn.

To say " no U is ^, during F" is evidently the same
as to say " all Uis F, during F; " that is,

so F10 — Fu ,

Since every proposition with zero as one or both of the
suffices is thus expressible in a form with no suffix equal
to zero, each suffix used will be supposed greater than
zero. The suffices u, v are also supposed less than U, F",
just as u was supposed less than U in the preceding
section. Faji will sometimes be used to include all six
of the fundamental propositions : that is, a will be con
sidered as having any one of the values 1, w, or u' ; ft as
having any one of the values 1, v, or v'.

ON A NEW ALGEBKA OF LOGIC.

For inference ly combination of such propositions we
have tup following simple rules, which are seen to be the
same as in § 1 : —

The conclusion from the
product of two premises is the
product of their predicates
affected by suffices which are
the products (in extension) of
the suffices of the premises.
Thus

When all the suffices are 1.
the relation between the pro
duct of the premises and the
conclusion is equality ; other
wise it is -<, — that is, impli
cation. Thus

FulGn-<(FG)ulJ
etc.

The conclusion from the
sum of two premises is the
sum of their predicates affected
by suffices which are the sums
(in intension) of the suffices
of the premises. Thus

When none of the suffices are
1, the relation between the
sum of the premises and the
conclusion is equality ; other
wise it is -<, — that is, impli
cation. Thus

Fuv + GUV=(F + G)m,

Flv+ Guv-< (F+ G)uv,

etc.

But, by an exception to the

rule, do not have FulGlv-< (FG\V since Glv is not of the

form (Gj)v.

These formulas really follow at once from those in § 1.
Thus Fn may be written (^1)1 ; hence by § 1 we have

and

(Fu)v + (Gu)v = (Fu + Gu)v =((F+
So in general we have

and

(Fa+

O1S: A NEW ALGEBRA OF LOGIC. 91

the addition of the suffices being taken in the same sense
as in § 1 ; that is,

1 + 1 = 1, l + u' = u',

1 + U — U, U + U1 = Uj

u + u — u, u' + u' — u'j

with like equations for v, v'. The second set of equa

tions means

All of + same part of = same part of,

Some of + same part of = some part of,
Same part of -f- same part of — same part of,

and a -little consideration will show that the formulae
hold as well for the accented suffices as for the unac
cented.

The following formula is evident : —

(FG)afi ~< Fo.fi Ga.fi '

For inference by elimination we have only to consider
the general form Fa^ and the rule is precisely the same
as the rule for elimination given in § 1, viz. : Any set of
terms may be eliminated by erasure provided no aggregant
term is thereby destroyed. Thus

(a + bx + cdxy + ey)afi -< (a + b + cd + e)afi,

and the reason of the rule need not be repeated.

The rule for inference by predication is also evidently
the same as that previously given. Thus

(a + b + cd + e)ap -< (c -< a + b +
and, in general,

If, after the multiplication has been performed, mF= mP,

then we have

Fa -<(m-< P)a/3.

92 ON A NEW ALGEBRA OF LOGIC.

Since propositions of the form Fn can be multiplied
without loss of content, and propositions of the form Fuv
can be added without loss of content, the most general
proposition involving the six fundamental elements is of
the form

or H (S^u + 3Gul + SJGT^ + 2Klv + 2Llv + Muv),

where F, 6r, etc. are logical polynomials of class terms.
But to the six elements just considered we may add as
elements the forms $x, <2>r considered at the close of § 1,
where $ is of the form Pu + 2 Q^ or PJIQU (see page
79) ; so that $1? <&v will be of the forms

(Pu+2Qi)i, WI ft,)..

It is clear that (PM + SQ^V = Puv + 5Qlv, and that
(^PinQu)l = PnnQul; but for the two forms of $M <PV
just given, no such reduction can be made. The suffices
within the parentheses of ^1 , $v refer to the universe of
class terms, those outside to the universe of time. If
the relative meaning of these suffices be reversed, so that
the suffices inside the parentheses refer to the universe
of class terms and those outside to the universe of time,
we have two other prepositional elements. Thus in
order to distinguish the meaning of the suffices clearly,
it will be necessary to use the capital letters Z7, F", and
write the four forms just considered as

or, in full,

(Pu + 2Qv)r, (P^nft,),, (pv

The negative of <fry is $r, which is of the form ¥*„. So the
negative of Xv is Xu, which is of the form flu. As ex
amples of £r, X^, suppose the universe of class terms
to be plane figures a, 6, etc., on a blackboard, and the

ON A NEW ALGEBRA OF LOGIC. 93

universe of time to be an hour. Let P = cib, and
Q = c + d ; then

means " during every part of the hour, either some a
is b, or no c is d" while

means " for every part of the blackboard, it is true that
it is either sometimes both a and b, or never both c
and d" So, as examples of ¥"„ and flu we have, re
spectively,

\(ab)oQ + &)„}„

which means " at some time during the hour, all the
blackboard is a&, and some of it is c + d" and

which means " some part of the blackboard is always
ab and sometimes c + c?."

Adding the four prepositional elements just described
to the six described previously, we see that the most
general proposition is of the form

To illustrate the method of inference from propositions
like the foregoing, consider the solution of the following
problem : —

Six plane figures, a, b, c, d, e, f, on a blackboard are
constantly changing their size, shape, and position during
an hour under the following restrictions : —

I. The area of c and d together is always included in
the area of a and b together, or else, during a certain
portion of the hour, e is equal to the part common to d
and f.

94 ON A NEW ALGEBKA OF LOGIC.

II. The part of a which is not e is always included
under the part common to d and f which is not b, or else,
during the whole hour, it is true for some part of the
board that all b is both c and e.

III. Either a and d are non-existent and e always
covers the board, or else it is always covered either by b
or by c.

What may be inferred (1) about the relation among
a, c, e and f, independent of b and d; (2) about the re
lation among a, c, e, independent of b, d, f ?

The premises are

I. (a + b + cd)n + (def + de + ef\v,

II. (d + e + ldf)n + (5 + ce)ul

III. ade

From the product of the first two we infer

(ab + acd + ae + be + cde + a5df)n + (def + ade + aej)
+ (al + led + ace + bce)ul + (tdef+ Me +

and multiplying this proposition by the third premise
according to the preceding rules, we get as an inference

(Me + acae)n + (abode -\- dbcde)ul + (bdef + abcle + abef)w
+ (cdef+ acde + acef)lv, + (ale + ace + bce)ul + (cdef+ Me
+ lcej)uv + {(ab + be)u+ (abc + ace + bee + alcdf)L}y,

three of the complex elements reducing to simple ones
according to the formulae,

G)uv.
Dropping b and d from the above proposition, we get

(ae)n + (ae)ul + (ae + ef)w + (ace + cef)lv, + (ac + ce)ul
+ (ce + cf)w + {(a + e)v + (ac + ce + cf)^}^

ON A NEW ALGEBRA OF LOGIC. 95

But in a sum, any term may be dropped which implies,
or is included under, another term.

O)n -< («e)Bl and (ace + cef)lv, -< (ae + ef)w;
therefore the above reduces to

(ae)ul + (ae + ef)lv, + (ac + ce)ul + (ce + cf)m + {(a + e)L
+ (ac + ce + cf)v}r,

which is the first quaesitnm, and may be read in words
" either it is always true that some e is not a ; or at a
particular part of the hour all a is e, and all e is /; or
during each part of the hour some c is either a or e ;
or at some part of the hour some c is either/ or not e;
or during each part of the hour either all a is e, or the
whole blackboard is c and all a is either e or/."
Dropping / from this result, we get

(a«)«i + (« + e)lv, + (ac + ce)ul+(c)uv + {(a + e)n+ (c)L}y.

But (ac + ce)ul -< (c\v and {(a + e)n + (c>)^-< (a + e)w
(<?)„„, therefore we get as the second qua3situm,

which means " either it is always true that some e is not
a ; or during some particular part of the hour all a is e ;
or there is sometimes some c." In like manner any
other set of terms can be eliminated by dropping them
from the product of the premises.

Propositions of more than two dimensions. If the
universe of relation be supposed to consist of three di
mensions, £7", V9 W, proceeding just as before we should
find that the number of fundamental propositions with
three suffices,

•^in > ^uio F«a.> Fuvw, etc.,
is twenty-six. The logic of such propositions is a " hyper "

96 ON" A NEW ALGEBRA OF LOGIC.

logic, somewhat analogous to the geometry of "hyper"
space. In the same way the logic of a universe of rela
tion of four or more dimensions could be considered.
The rules of inference would be exactly similar to those
already given.

Allusion has already been made to the fact that the
propositions considered in this and the preceding section
may be regarded as relative terms. In the first section,
the two fundamental propositions, Fl and Fu, are dual
relatives. Ft means "F is a description of every part of
U-, " and Fu means "F is a description of some part
of £7." Thus Fl and Fu correspond to the two funda
mental dual relatives. So in § 2, Fn is a triple relative
term, meaning 11F is a description of every part of U
during every part of V" Thus the six fundamental
propositions of two dimensions correspond exactly to the
six fundamental varieties of triple relatives, and so on.

§ 3. On Certain Other Methods.

The propositions A and 0 in Mr. Peirce's notation
are, respectively,

Y,
X^< Y.

Mr. McColl expresses them in a similar way, using a
different symbol for the copula. Both Mr. McColl and
Mr. Peirce have given algebraic methods in logic, in
which the terms of these propositions are allowed to
remain on both sides of the copula.

In the method of § 1 (of which § 2 is an extension),
the propositions A and 0 are expressed as follows : —

(X+ Y)19 equivalent to oo -< X + Y,

ON A. NEW ALGEBRA OF LOGIC. 97

that is, all the terms of the universal proposition are
transposed to the right hand side of the copula, while
those of the particular proposition are transposed to the
left-hand side.

If these propositions be expressed in the reverse way,
namely, —

the rules of inference become the exact logical negatives
of those in § 1, addition taking the place of multiplica
tion, arid vice versa. XY-<^ 0 is equivalent to (XY)0,
meaning "none of U is XY" as has already been ex
plained. GO ^<; X + .Fmay be represented by (X+ Y)q,
meaning " some of U is not X + Y" or " there is some
thing besides X+ Y." Thus jP0 and Fq are the two
fundamental forms of proposition in this method, arid
the rules of inference by combination are

F0G0=(F+G\
F,Gq-<(F+G)<

F+ G =

Fq+ GQ-< (FG)q
F0 + G0 -< (FG\.

Elimination is performed by multiplying together the
co-efficients of the quantities to be eliminated.

Boole's method, as simplified by Schroder, lias been
extended by Miss Ladd, in the foregoing paper, so as to
express particular propositions without the use of Boole's
objectionable " arbitrary " class symbol. She has ex
pressed A and 0 as follows : —

XY V, equivalent to XY -< 0,

XY\[, " " XY^Q.

Thus F0 and Fu are the two fundamental forms of propo-

98 ON A NEW ALGEBKA OF LOGIC.

sition in her method, and the rules of inference ly
combination are

FQGU-<(FG)U

FuGu-< oo.

Elimination from F0 is performed by multiplying co
efficients ; from jPM, by adding them.

One more method remains to be noticed, — the negative
of Miss Ladd's method, in which A and 0 are expressed

o>-< X+ Y,

and where F1 and Fq are thus the two fundamental forms
of proposition. The rules of inference ly combination are

Fq+Qq=(FGT)q

(F+G)q FQ+G,-<(FG\

and elimination from Fl is performed by addition of co
efficients ; from Fq, by multiplication of coefficients.

§ 4. On a special notation for De Morgan's Eight Propo
sitions, with an extension of the same to similar propo
sitions of three or more terms.

It is proposed in this section so to change the notation
previously given for De Morgan's eight propositions that
the elimination of the middle term will be performed by
an algebraic multiplication of the premises. Denote by
J', E', 0', A what J, E, 0, A become when each term
is replaced by its negative. The propositions J, E, 0, A,
and their complementaries I', E', 0', A have already
been represented (see page 76) respectively by

OK A NEW ALGEBRA OF LOGIC. 99

and also, since Fl — FQ, by

(ab)u, («5)o, (a5)«, (a&)0,
Let these be now chaned to

where the negative of a term is now denoted by affecting
it with the exponent— 1, and the negative of a propo
sition is denoted in the same way. Thus

(ab~l) means " some a is not b"
(ab-1)-1 " "all a is 6," etc.
With this notation there is the following simple

RULE OP INFERENCE. Excluding products of two par
ticulars, the conclusion from a set of premises is their
algebraic product, with the convention that the appearance
of a middle term in the result indicates that there is no
conclusion.

Thus, Barbara is

-1)-1 x (sm-1)-1 -< (sp-l)~\

and Darii is

(mp~l)-lx (sm)-< (sp)-,

but from A and 0 as premises we get

(rap-1)"1 X (sra-1) -< oo,

the middle term not disappearing from the product.

From the nature of this notation, just as with that of
§ 1, the order in which the two terms of a proposition are
written is indifferent, and consequently the figure of a
syllogism -is indifferent. Thus, (mp) is the same as
(pm). Thus Celarent and Cesar e are

(mp)~l x (sm-1)-1 -< (sp)~\
Darii and Datisi are

(mp-l)~l X (sm) -< (sp).

100

ON A NEW ALGEBRA OF LOGIC.

Ferio, Festino, Ferison, and Fresison are
(mp)-1 X (sm) -< (sp-1).
Camestres and Camenes are
-1 X

Baroko is

(pm~l)-} X

1 -< (sp)-\
-< (sp-1.

Bokardo is

Disamis and Dimaris are

X (ras"1)"1 -< (sp).

X (ms-1)-1 -< (sp-1).

This rule of inference is seen to accord with the now-
recognized invalidity of the moods Darapti, Felapton, and
Fesapo. Thus the premises of Darapti are

(mp~l)~l x

from the product of which m does not disappear, and
there is therefore, according to the rule, no inference.
The same is true for Felapton and Fesapo. The premi
ses of Bramantip are

(pm~1)-1 X (ms~l)~l, which -< (s"1^)"1.

The following Table gives all the valid moods from
De Morgan's eight propositions : —

-lm^) (prn)

(sm)'1
(sm-1)
(s-%)

(sm)

(sp)

(9)

to)

ON A NEW ALGEBRA OF LOGIC. 101

There are twenty-four valid moods, but if no distinction
be made between s and p, these reduce to the twelve in
either half of the Table, the Table dividing itself sym
metrically along the diagonal from left down to right.
The unsymmetry of the Aristotelian system is seen from
the fact that the fifteen valid moods of the Aristotelian
system comprise only eight out of the twenty-four of the
Table, and these eight select themselves very unsym-
metrically, being those underscored by dotted lines.
From the three formulae

~l -< (sp)-\
(sm) X (pm)-1 -< (sp~l),
(sm)~l X (pm) -< (s-»,

the whole twenty-four syllogisms of the Table may be
obtained by substituting for m, s, and p their negatives
in all possible ways, each formula yielding eight.

Mr. Hugh McColl, in his papers on logic in the " Pro
ceedings of the London Mathematical Society " (Vol. IX,
et. seq.), has been using a notation for the copula identi
cal in meaning with that of Mr. Peirce. He uses a colon
to denote implication, instead of -<. Mr. Peirce has
recently told me that Mr. McColl justifies his use of the
colon by its mathematical meaning as a sign of division.
Thus Barbara and Celarent are

m : p m : p

s : m s : m

.'. s :p /. s :p,

and the analogy to division is obvious. But this analogy

102 ON A NEW ALGEBRA OF LOGIC.

exists only in the two universal moods of the first figure.
Thus Cesare and Festino are

p : m p : m

s:m s -7- m

.-. s :p s+p,

where -f- is the negative copula, and the analogy to
division is wanting. In the notation of this section the
analogy of the premises to ratios, and of the conclusion
to their product is more nearly complete.

Extension of the preceding.

Let (abc) denote " a, b, c have something in common,"

and (abc)~l " a, b, c " nothing " "

By substituting for a, 5, c their negatives in all possible
ways, we get sixteen propositions concerning three terms,
thus seen to be analogous to De Morgan's eight concern
ing two terms. In the same way we may get thirty-two
propositions concerning four terms, and 2.2n propositions
concerning n terms. The formulas of inference from
propositions like the above are

(ab...ffh...t) (h...lm...q)~l-< (ab...g) (m...q)~\
(ab Id) (l-lm q)~l-< (ab km...q)~l.

In the first, where one premise is particular, inference
can take place independently of any number of middle
terms, provided each term is positive in both premises,
or negative in both. In the second formula, when both
premises are universal, inference can take place inde
pendently of only one middle term, and this must be of
different quality in the two premises. By an obvious sub
stitution these two formulae are reduced to the formulae

ON A NEW ALGEBRA OF LOGIC. 103

previously given involving only two terms in each pre
mise. Thus

(r1*)-1 -< (a*)-1.

That is, the premises of the first mean " that which is
common (x) to a, 5, ...#, has something in common
with the common part (#) of A, . . . Z ; " and " the common
part (j/) of h, . . . I has nothing in common with m, . . . q"
Whence the inference is (xy~l), or (ab . . .g) (m. . . q)~l.
The premises of the second mean " whatever may be
common (x) to «, b, ... Jc, has nothing in common with
Z; " and " whatever may be common (z) to m, . . . q, has
nothing in common with non-Z." Whence the inference
is (xz)~l, or (ab . . . km . . . q)~\

(abc) means (ab) (ac) (be),
/. (abc)'1 " (ab)-1 + (ac)-1 + (bc)~\

Thus any one of these propositions is reducible to a
function of De Morgan's eight.

§ 5. Note on De Morgan's Twenty Propositions.1

It is proposed in this section to consider a simple
method of deriving and writing De Morgan's Twenty
Propositions. Let A = all of A, a — part of A, A = all
of non-J., and d = part of non-J., where part of is under
stood to mean less than the whole of. Let a second term
B be modified in the same way. Then, by affirming
and denying identity between each modification of the
first term and each modification of the second, we get
thirty-two propositions, of which, however, twelve are
duplicates. That is, the process yields twenty distinct

1 See his "Syllabus of Logic," §§ 24-62.

104

ON A NEW ALGEBRA OF LOGIC.

propositions, and they are easily seen to be the twenty
of De Morgan. Let the affirmation of identity between
two terms be denoted by their juxtaposition, and let the
denial of the same be denoted by a line extending over
both terms. Then we have the following

TABLE OF DE MORGAN'S TWENTY PROPOSITIONS.

AB, or AB

Ab, " aB

Ab, « aB

~Ab, " ^§

Ab, " aB

aB, " Jb

aB, " AB

aB, " Tb

~OB, " A5

~ab

Thus, AB means " the whole of A is identical with the
whole of j£." It is obvious that AB is equivalent in
meaning to AB. The second proposition, Ab, means
" the whole of A is identical with a part of B" (that is,
all_ A is B, and some B is not A). It is clear that
aB, or " a part of non-A is identical with the whole of
non-,5," is the same as Ab. To take an example from
the other side of the Table, AB means " it is not true
that the whole of A is identical with the whole of B."
This is simply the denial of the proposition AB. Ab
means " it is riot true that the whole of A is identical
with a part of 5," a simple denial of Al.

The propositions below the horizontal line of division,
which are differentiated from those above the line by
containing only small letters in their symbols, are De
Morgan's eight " simple " propositions.

ON A NEW ALGEBRA OF LOGIC. 105

(A part of A is a part of B } £

~ (Some A is B >

— ( It is not true that a part of A is a part of B \ ™

~~ I ~vr A ' 7? i

- _ ( A part of A is a part of non-B ) Q

\ Some A is not B )

-Y ( It is not true that a part of A is a part of non-B ) ,
= lAll.iis2? >'

The remaining four of these eight are derived from these
four by the negation of their terms. This notation for
the eight propositions differs only slightly from that
employed in previous sections.

De Morgan derived his eight " simple " propositions
by applying the Aristotelian forms A,E,I, 0 to the four
pairs of terms X, Y\ X,Y\ X,Y; X, Y. This process
gives sixteen propositions, of which eight are duplicates.
The other twelve of the twenty he called "complex,"
because they are compounded of the eight simple propo
sitions, as follows : —

~al)Xtib AB = al + ab

Ab — aft X ab Ab — ab + ab

aB = aL X ab a,B = aB + ab

~AB — d6 X ab AB = at + ab

~Ab — ~ab X ab Ab = ab + ab

aB = ab X ab ~aH = at + ab

The following Table gives the conclusions from one
hundred out of the possible four hundred combinations
of two premises from this system of twenty propositions :

106 ON A NEW ALGEBEA OF LOGIC.

PM Pm pM pm pm PM Pm pM pm pm

Sm
sM
sm
sm
SM
S^
^M
sm
sm

Sp
7P

By applying the sign of negation first to the S, then to
the P, then to both the S and the P, the remaining three
hundred are obtained. According to De Morgan, who
postulates that every term and its negative is greater
than zero, there are two conclusions not given in the
Table, namely : —

sm X pin -< sp,

sm X pm -< sp,

and from these are obtained six others by applying the
sign of negation to s and p. But according to the
definitions of Mr. Peirce and others, already alluded to,
these are invalid conclusions ; since, being particular,
they imply the existence of their subjects, while the
universal premises do not.

OPERATIONS IN RELATIVE NUMBER WITH
APPLICATIONS TO THE THEORY OF PROBA
BILITIES.

BY B. I. OILMAN.

THE purpose of this Paper is to deduce the formulae
for the addition and multiplication of Relative Number,
and to apply them in demonstrating the well-known
fundamental theorems of Probabilities, according to Mr.
Peirce's method of dealing with the subject.

If a relation be that which we perceive when a group
of objects are viewed together, but which we do not
perceive when we regard each separately, then any act
of comparison will bring to view a relation. If the
objects compared are two in number, the relation may
be called a dual one.

Such a dual relation may be viewed in two lights, or
we may say it splits into two elementary forms, accord
ing as one or the other object is our starting-point in
comparing the couple. The two are called the direct
relation and its converse. Thus, what is ordinarily
termed a relation may be said to have ends, being based
on a comparison having a direction. One of these ends
is called the relate, the other the correlate.

A relative number is a number obtained in either of
the two following ways : first, by dividing the number

108 OPERATIONS IN EELATIYE NUMBER.

of instances in which a given relation has a relate in a
certain class of objects by the number of objects in the
class ; or, second, by dividing the number of instances
in which a given relation has a correlate in the given
class by the number of objects in the class. Hence, for
a given relation p' we have two such relative or aver
age numbers, — one, the number of instances in which
p' has a relate of the class ?/, divided by the number
of ?/'s ; and the other, the average number per y of p'
whose correlates are ?/'s. The former might be called
the relate-number of p', the latter its correlate-number.
But if we extend the class y to include all the objects
in the universe, since the number of instances in which
the relation p' occurs having a relate which is an object
in the universe, is equal to the total number of times
p' occurs at all, and the same thing is true of the number
of occurrences in which it has a correlate which is in the
universe : it follows that for both relate and correlate
numbers we get the average number of relations p' per
object in the universe. That is, any relation p' has but
one (what we shall call) general relative number.

Denoting each object in the universe by a certain
letter, each possible different couple of objects (con
sidering those couples as different in which the same
elements occur in a different order) will be symbolized
once, and only once, in Mr. Peirce's scheme of pairs,
as follows: —

A:A A:B A:C A:D ...

B:A B:B B:C B:D ...

C:A C:B C:C C:D ...

D:A D:B D:C D:D

OPERATIONS IN RELATIVE NUMBER, 109

Now if in this scheme of pairs we assume the relation-
direction to be constant, say from left to right, — that
is, that the right-hand members of the pairs are the cor
relates, — it will follow that any single instance of any
relation must subsist between some one, and only one,
of the pairs. Marking in any way, as by a circum
scribed circle, those pairs between the components of
which subsists the relation p' ; and marking by a circum
scribed square instances of the relation p", — we shall
have in general some pairs surrounded by circles, some
by squares, and some by both.

Whence if p' and p" denote respectively the number
of individual relations comprised in the general relations
p' and p", we shall have

p' -f p" = number of pairs surrounded by circle alone + num
ber of pairs surrounded by square alone + twice
the number of pairs surrounded by both circle and
square = p',p" + P",p' + 2 p', P"

in which p', p" denotes the number of pairs concerning
each of which it can be said that it is in both the rela
tions p' and p" ; and p', p" denotes the number of pairs
which arc . at once in the relation p1 and not in the
relation p". Again,

p' -f p" = number of pairs in circle, or square, or both + num
ber in both = (p1 -I- p") + p1, p",

in which— according to Mr. Jevons's notation — (p1 -|-p")
denotes that class of pairs concerning each member of
which it can be said that it is either an instance of p' or
of p" or of both. Now, since a general relative number
is the total number of individual instances of a relation,
divided by the number of objects in the universe, if we
indicate the number of objects in the universe by oo,
-^ will indicate the general relative number of the rela-

110 OPERATIONS IN RELATIVE NUMBER.

tion pr. Symbolizing this quotient by [y ] , and dividing
both sides of the above equations by oo, we have

[p' ] + [P"] = IP', P"-] + [A p'] + 2 |y, p"] = [y .,. p'/] + [y, p"].

We thus have reached two formulae for the addition
of two relative numbers. Similarly, we have for the
addition of three relative numbers

0'] + [P"] + [>'"] - CP', ?', P'"] + [P", P', P'"] + [P'", P',P"]
+ 2 [P', p", p"'] + 2 |y , p'", p//] + 2 [p'/, p'", p/]

- [p' •!• P" -i- P/;/] + CP', P", P'"] + iy, p"f, P"]

+ [p^P//',P/] + 2[P',p^^/],

Similar formulae may be deduced for the addition of n
relative numbers, as follows : —

Pn-2])
Pre-3])

= [p'-|-p"-|Y" - ... .j.p»]

+ LP>p">p"' •••?"]+...+ [p""1^*?? • • • p'l~2]

This latter formula gives, when the relations are
mutually incompatible, — that is, when no two of them
can subsist between the same pair, — a much simpler
result : —

CP;] + [P"] + . . . + [p"] = [p' .|.p".|.p'" . .
all the other terms reducing to zero.

OPERATIONS IN RELATIVE NUMBER. Ill

To obtain a formula for the multiplication of relative,
numbers we notice that

Let x, which may be any number, signify the number
of different existing groups of three objects, such that
the first is to the second in the relation p' and the second
to the third in the relation p". Such a group may be
called a relative sequence, and may be denoted by prp"
without the comma. Then

If now

the formula becomes

In this case, therefore, the product of the relative
numbers of the two given relations equals the relative
number of the sequence formed from them.

Multiplying numerator and denominator of — by the

00 *

number of objects in the universe, it becomes p X2°°.

The numerator of this fraction is a number equal to the
number of different triplets obtained by combining each
p' with every object in the universe. Between the second
and third members of these triplets either the relation
p' or p1 must hold ; and no relative sequence of the form
p'p" or p'pn can exist which does not appear among them.
Hence the number p' x oo equals the sum of the num
bers of p'p" and p'p". The denominator being the square
of the number of objects in the universe is equal to the

112 OPERATIONS IN RELATIVE NUMBER.

number of possible pairs, and each of these is either
pn or p

00
and

P'P" _ P'P"

That is, the average number of sequences pfpn per
each p" is the same as the average number of sequences
p'pn per each p'f. Hence, whether the relations in which
any given individual stands to the others in the uni
verse are all p", or one or more p" and the rest pn, will
make no difference on the average in the number of
relative sequences whose first member is p' of which it
is the intermediary. The number of such sequences in
the case of any individual being the number of the ob
jects standing to it in the relation p' multiplied by the
number of objects in the universe, it follows that the
number of objects standing to any given individual in
the relation p' is not affected by the circumstance of its
being p" to one or more objects.

P'P" P' j. P'P" P" i

Similarly, from ^- = — we may get *--~- = — , whence

eZ = p'p" + pp' or ^ = ££ ; that is, whether an object
P p+p P p'

is correlate in any relations p' or not, will make no
difference on the average in the number of p"s of which
it is the relate.

For instance, letting p' indicate the relation borrower
from, and p" the relation trustee of, this condition ex
presses, first, the fact that a man's being a trustee makes
no difference on the average in the number of borrowers

OPERATIONS IN" RELATIVE NUMBER. 113

from him ; and, second, that a man's being a lender or
not makes no difference on the average in the number
of funds which he controls as trustee. Such relations,
from one of which nothing can be inferred regarding
the presence of the other, are called independent re
lations. Hence for independent relations,

!>'] x

The expression p'p" here denoting the number of
relative sequences of that form, if we define a compound
relation to be a combination of such relative sequences
as have the same individual object as relate, ', and also
the same individual object as correlate, f/, we shall have
each compound relation consisting of as many sequences
as it has intermediary objects. Hence, in order to ex
press the number of p'p'f$ in terms of compound relations
of that form, to the total number of compound rela
tions we shall have to add the number of those which
have two intermediaries, since they each contribute an
extra sequence ; and to this sum we must further add
twice the number of compound relations having three in
termediaries, three times those having four, etc. Hence
we have for the number of relative sequences expressed
in terms of compound relations,

= P'P" + P'P" + 2 P'P" + ...(n-l) P'P

wherein P'P" denotes the total number of compound
relations of the form p'p'1 having whatever number of
intermediaries; P'P" denotes the number of such com
pound relations having two intermediaries, etc. Whence,
dividing through by oo, we have

... (n -

114 OPERATIONS IN RELATIVE NUMBER.

and the following formula results for the multiplication
of independent relative numbers : —

- 2 3

CP'] [P"] = [^P"j + \_PiP"-] + 2 [P'P//] . . . (n - 1) [PP"].
By a somewhat different and a longer process of proof,
it can be shown that for independent relations the follow
ing formula holds for the multiplication of n relative
numbers : —

2[P' . . .

Here it is to be noted that the superscribed numbers do
not refer to the number of intermediaries, but to the de
gree of connection, the number of ways in which relate '
and correlate n are connected by chains of relation.

The continued product of the numbers indicating the
simultaneous intermediaries at the successive steps, it
is easily seen, cannot be less than r nor greater than
r(n~l\ when the connection in the given relation is an
r-fold one. Since permuting the multipliers does not
change the left-hand member, the right-hand member
remains constant in whatever order the elementary rela
tives are compounded.

Through the addition formula we have reached what
we may call polynomial relative numbers, of the form
[p! -I- p" -I ..... |. pM] which expresses the relative number
of that class of pairs, each one of which is an instance
of some one or more of the relations p'. . . pn. In the
case of incompatible relations we have the equation

[p'-lV'-l ..... |-PB]

Whence the multiplication of polynomial relative num
bers reduces in the case of incompatible relations to that
of monomials.

OPERATIONS IN RELATIVE NUMBER. 115

The involution of a monomial relative number gives
the ordinary result of multiplication, except that all the
elements of the resulting compound relation are the
same. If we involve an incompatible polynomial, we
shall get a result according to the multinomial theorem,
consisting of monomial powers and products.

In order to apply these results to the theory of proba
bilities, we shall require to make a supposition in regard
to the character of the relations we are to consider. If
a relation is perceived whenever we compare objects, it
follows that a relation will be noticed when we think
of an object as existing at successive times; for this
involves a comparison between its aspect at one time
and at another.

This relation between objects which differ, so far as
we see, only in existing at different times, we call iden
tity. The pairs in the principal diagonal of the relative
scheme exist in this relation only, since what we call the
same or an identical object is both correlate and relate.

The relative number of the relation of identity is evi
dently unity, since it occurs once, and no more, for every
individual in the universe. Now we can, if we please,
agree to bring the various individual relations, — that is,
relations subsisting between individual objects, — which
together make up the total extension of the general re
lation identity, into various classes according to the
character of the objects they identify. This will create
as many kinds of relation of identity as there are classes
of objects in the universe, and their relative numbers will
vary from - up to unity, and will express the propor
tion of objects of the different kinds in the universe.

Further, we may agree to take for the divisor of our
relative number, for our y, instead of all the objects in
the universe, some limited portion of them, say the class

116 OPERATIONS IN RELATIVE NUMBER.

b. This will be a return to the special relative number
mentioned at the beginning of the paper ; but it is evi
dent that since the relation whose relative number we
seek is a relation of identity, every instance of it which
has its relate in the class b will also have its correlate
in that class, and vice versa; so that the relate and cor
relate number of the relation will be the same, and may
be called simply its relative number. Such a relative
number will mean the number of identity relations of
the form a to be found among the relations pertaining
to the individuals of the class b divided by the number
of those individuals; that. is, the number of a's among
the 6's, divided by the number of &'s, or, in other words,
the proportion of the genus b that is of the species a.
If we regard events as the objects between which the
relations we are considering subsist, an identical relative
number will express the proportion in which a certain
species of event exists in a genus. With this ratio will
vary the expectation with which we shall look to see a
case of the genus a case also of the species ; it may be
said to measure the value of the genus as a proof of the
species, — to measure, that is, the prove-ability, or proba
bility, of the species from the standpoint of the genus.

On this view of probability it has to do, not with
individual events, but with classes of events ; and not
with one class, but with a pair of classes, — the one
containing, the other contained. The latter being the
one with which we are principally concerned, we speak,
by an ellipsis, of its probability without mentioning the
containing class; but in reality probability is a ratio,
and to define it we must have both correlates given.

An identical relative number, then, when the identities
considered are events, will be the ratio of a specific to a
generic occurrence ; and this ratio is called the proba-

OPERATIONS IN RELATIVE NUMBER. 117

bility of the species with respect to the genus. The
mathematical combination of probabilities will therefore
take place in accordance with the formula for relative
number already reached, with such modifications as re
sult from their application to relations of identity.

In establishing by these formula the fundamental the
orems of probabilities, let the individuals in the uni
verse we are considering be events ; and let a denote a
certain kind of relation of identity between them, — that
is, a certain class of events, — and a the remaining rela
tions of identity, that is, all the rest of the events in the
universe. The general relative numbers of a and a —
that is, the general probabilities of a and a in the uni
verse — will be denoted by [a] and [#].

From the addition formula we have

[>] + [«] = [a.j.a] + [>,«]•

The first term of the right-hand member is the relative
number of that class of pairs, each of which exhibits
either or both of the relations a and a ; and the second
term of the right-hand member is the relative number
of that class of pairs, each of which exhibits both the
relations a and a. But since by definition a is a part
and a the rest of the existing relations of identity, no
event exhibits them both, and [a ,a] — 0 ; while the num
ber of relations a .|. a equals oo, and hence [a .\. a] = 1.
Thus we have

M + [«]=!

[5] =!-[>] (1)

or, the probability of the negative of an event equals
unity minus the probability of the event.

The relations a and a are incompatible relations ; that
is, they cannot subsist at once between the same pair.

Incompatibility means, therefore, in the case of rela-

118 OPERATIONS IN RELATIVE NUMBER.

tions of identity between events, that no one event can
be of both species ; the species are mutually exclusive,
- the events, as we say, cannot happen together. Such
events may be called exclusives, and we may denote
by the term alternatives specific events which together
make up a genus; that is, exclusives one or other of
which must happen if the generic event happen at all.
The generic event consisting of the occurrence of any
one of a number of exclusives may be called an alter
nating event.

The abridged form of the addition formula, when the
relations are incompatible, gives the following as the
probability of an alternating event : —

[> -I- * -I- <H I- "] = M + p] + [c] + . . . + W (2)

That is, the probability of an alternating event is equal
to the sum of the probabilities of the exclusives of which
it is composed.

The expression a , b ,c , cl . . . n denotes an event which is
at once a, 5,0, note?... and not n\ and [a,b,c,d. . .n~\
denotes the probability of such a compound event. If we
have certain events of known probability, a,b,c . . . n
which are not exclusives, and wish to obtain the proba
bility of the occurrence of some one, and only one, of
them, the desired expression reduces to a sum of such
compound probabilities. For the event in question will
be either (a, 5, . . . w,w), or (a, 6 . . . m,w), etc.,
or (a, 5 . . . m,ri) ; and these compounds being mutu
ally exclusive, the event is an alternating one, and its
probability is expressed as follows : —

[a,5 . . . n.\.a,b . . . n.\. . ..\.a . . . m,n~\ =

[a,l . . . n] + \_a,b . . . ri] + . . . + [a . . . m,w]

This result being in terms of the probability of compound

OPERATIONS IN RELATIVE NUMBER. 119

events, to make it available we must have means of cal
culating compound probabilities from simple ones.

The formula obtained above for multiplying relative
numbers expresses the result of such a multiplication in
terms of the relative numbers of compound relations.
In the case of identical relations, these would be com
pound relations of identity. But since no object or
event is in the relation of identity to more than one ob
ject or event, — that is, itself, — each compound relation
of identity must consist of a single relative sequence;
accordingly all the terms after the first in the right-hand
member of the multiplication formula disappear, the re
maining term being the relative number of a relation
of identity compounded of all the multiplied factors.
But since all the objects concerned in this compound
relation from relate ' to correlate n are one and the same,
it is no longer a sequence of relations, but a coexistence
of special identities, — a coexistence of characters ; and
its relative number is the relative number of such co
existences, — of objects or events in which coexist all
the given special identities that belong at once to all the
given species. The condition that the relations should
be independent, that is, that between any two of them,

a, b a,T)

T T

for relations of identity becomes the condition that the
proportion of 5's that are also a's should equal the pro
portion of 5's that are also o's ; in other words, that an
event is b should make it neither more nor less likely
that it is also a case of a, and vice versa.

We thus see that the multiplication of identical rela
tive numbers, when the relations are independent, will
give the relative number of the events in which all the
multiplied identities coexist. The probability of a com-

120 OPERATIONS IN RELATIVE NUMBER.

pound event, therefore, when the components are inde
pendent, may be found by multiplying together the prob
abilities of all the components. Applying this principle
to the case of the compound events

[a, I ... u] + \_a,b, ... n\ ... + [«... m,n\,
we have for the probability of the occurrence of one, and
only one, of n independent non-exclusive events,
[a, I . . .n.\.a,b,c . . . n.\ ..... \.a . . . m,ri]
= M P] [>]•••[>] + [a] [6]...[w]...+ [«]...[m] [>]. (3)
For the probability of the occurrence of some one or
more of n independent non-exclusive events, we obtain
by transposition from the second form of the general
addition formula, —

& -I- * '!• c -I ..... |- n] = [a] + [£] + . . . . + [>]
- H M P] . . [n\ -...-[«]... [m] [n]
-2[a]J?][c][<Z]...[n]-. . .-2[«]...[q [m] [w]

-(»-!) M[ft][c]. . . M (4)

Since the probability of a compound event is the pro
duct of the probabilities of the components (when inde
pendent), we have the following equation : —

[a,M...n]== M [&] [c] . . . M (5)

which gives us

that is, the probability of any event is equal to the
probability of any compound event into which it enters,
divided by the probability of the compound event made
up of the remaining components.

OPERATIONS IN RELATIVE NUMBER. 121

"We may obtain an expression for the probability of
a compound event when the components are not inde
pendent, by noticing that in establishing the formula
for multiplication the independence of the relations ena
bled us to substitute in the left-hand member of the

equation, -^ for ^—-. If the relations are not independ
ent, this is not permissible ; whence indicating ^~- by
\_p'p"]P" the equation reads

ov '>[>"]

or for identical relations

[»,»]» [»] = [«,»],

in which [&,6]& denotes the proportion of a, 6' s among
5's, the probability that an event of the genus b will also
be of the species a. An extension of these considera
tions gives the general formula

[>,&.. .w]6...B [&,c.ra]c...» [c,d..ri]dtttn...\m9n]n [ri] =

O,a...rc]; (6)

that is, the probability of a compound event, when the
components are not independent, is equal to the general
probability of any one of the components multiplied by
the probability that one of the other components will
happen when the first happens, and so on until all the
components are exhausted.

Let us suppose that the compound event, instead of
being composed of n different events, is composed of
n like events, a. If these different occurrences of a are
independent, — that is, if the fact that a has occurred
once, makes it neither more nor less likely that it will
occur again, — we have

M = W (7)

122 OPERATIONS IN RELATIVE NUMBER.

While the mere fact that a has occurred will not,
contrary to the popular notion, make it any more or less
likely to recur, it is evident that in many instances at
tendant circumstances, as in the case of habit, may de
stroy the independence of successive occurrences.

If a is a compound of independent relations of identity,
as &, 6, c, ... m, the formula becomes

[(a,ft,c. •• m)M] = KM- • -mT

= (M P] M - - - W)-

= M»P]»[c]»...[m]»; (8)

that is, the probability of the repetition of a compound
event n times is equal to the product of the nih powers of
the probabilities of its components.

We have seen that a polynomial relative number ex
presses the probability of the occurrence of some one
or more of the separate events symbolized therein. If
the events are exclusives, it expresses the probability of
the occurrence of some one of them.

Considering two exclusives, a and 5, in order to ob
tain the probability that one or other of them should
occur n times, it is to be noticed first that this event
itself is not a single compound event, but a compound
alternating event, consisting of as many compound alter
natives as there are different arrangements of a and I in
n occurrences. Since the probability of an alternating
event is the sum of the probabilities of the alternatives,
the probability we seek will be the sum of the probabili
ties of all the compound alternatives ; that is, the sum
of all the products obtained by forming all possible
arrangements of n simple probabilities, each of which
must be either [a] or [b~\. In other words, the opera
tion of finding the probability of the occurrence of one dr
other of two exclusives n times, is the same as that of

OPERATIONS IN RELATIVE NUMBER. 123

raising the binomial [a] + [5] to the nth power. This
is otherwise seen thus : Since a and b are exclusives, —

but

Similarly, for more than two exclusives, the probability
of one or other happening p times is equal to the sum
of the probabilities of the exclusives raised to the p^
power, or

[(<H-H<H ..... I-")*] = (M + P] + W • • • + M)p. (9)
It may be observed in relation to the probabilities of
the compound alternatives of which these sums are made
up, that any one will be equal to all the others in which
the elementary exclusives enter in the same proportions,
although in different orders. The case of highest proba
bility will evidently be that consisting entirely of that
one of the elementary exclusives which has the highest
probability, and the case of lowest probability will be
that in which the elementary exclusive having the lowest
probability alone appears. On the contrary, other con
siderations show that the most probable proportions in
which different alternatives will enter into a series of
trials will be the ratios of their probabilities, while the
most improbable proportions will be those exhibited by
series consisting entirely of some one of the alternatives.
The same thing is true of exclusives ; the most probable
proportion in which they will be found in a series of
trials being the ratios of their probabilities. But while
with alternatives the sum of the probabilities of all
possible orders will continue to be unity, however the
number of trials is increased, with exclusives the sum
of these probabilities will decrease in geometrical pro
gression as the trials are repeated.

124 OPERATIONS IN RELATIVE NUMBER.

The results thus far reached, readily lead to other com
binations of probabilities, as in the following examples :
The probability of the occurrence of at least one of two
events with a third is given by the equation

(10)

in which, as in general in probabilities, the events are
supposed to be independent.

When a and b are exclusives, the same probability is
equal to

([a] + [J]) [«].

For any number of exclusives, and any number of
other events, the equation becomes

[OH-0-1 ..... 1-"), a, ft, . . • n\ =

For the probability of the occurrence of one, and only
one, of any number of non-exclusive events with any
number of others, we have

[(a,/3 . . . v .|. . .|. ... .|.a, . . . p.,v)a,b. . . n] =

[a] [J] . . . W( [a] [ft... [v]+ . . . + [a] . . . [£][>]) (12)

The probability that a will occur m times to n occur
rences of 5, — that is, that m a's will happen while n 6's
are happening, — will be the probability of the compound
event consisting of m a's and n 6's. The probability
that m a's will be succeeded by n 6's is [a]m[£]n, and the
number of different arrangements of m + n objects, m

of one kind and n of another, is ' ,' , ; whence the total

\m [n

probability is

If a and b were alternating events, this expression
would give the probability of the occurrence of some one

OPERATIONS IN RELATIVE NUMBER. 125

or other of TT exclusives m times, while some one or
other of p exclusives is happening n times. Substituting
the values assumed in this case by [a]m and [6Jn,we
have for this probability

+[/?]+...+ wr (M +[»]•• •+ !>])"• a*)

In this investigation of some modes of combining
probabilities, suggested by the consideration of Relative
Number, we have used the Addition formula in reaching
(1) the probability of negative events, (2) of some one
of n exclusives, (3) of some one, and only one, of n
non-exclusives, and (4) of at least one of n non-exclu-
sives. From the Multiplication formula we have ob
tained the probability of a compound event when the
components are either (5) independent, or (6) depend
ent ; and by a reference to the involution of Relative
Number have established formula for the probability
of the repetition of (7) simple (8) compound or (9)
alternating events. These results have been combined
in the more complicated cases (10 - 14) last considered.

A THEOEY OF PROBABLE INFERENCE.

BY C. S. PEIRCE.

THE following is an example of the simplest kind of
probable inference : —

About two per cent of persons wounded in the liver recover ;
This man has been wounded in the liver;
Therefore, there are two chances out of a hundred that he
will recover.

Compare this with the simplest of syllogisms, say the
following : —

Every man dies ;

Enoch was a man ;

Hence, Enoch must have died.

The latter argument consists in the application of a
general rule to a particular case. The former applies to
a particular case a rule not absolutely universal, but sub
ject to a known proportion of exceptions. Both may
alike be termed deductions, because they bring informa
tion about the uniform or usual course of things to bear
upon the solution of special questions ; and the probable
argument may approximate indefinitely to demonstration
as the ratio named in the first premise approaches to
unity or to zero.

A THEORY OF PROBABLE INFERENCE. 127

Let us set forth the general formula of the two kinds
of inference in the manner of formal logic.

FORM I.
Singular Syllogism in Barbara,.

Every M is a P;
S is an M ;
Hence, S is a P.

FORM II.
Simple Probable Deduction.

The proportion p of the M'a are P's ;

S is an M ;

It follows, with probability p, that S is a P.

It is to be observed that the ratio p need not be exactly
specified. We may reason from the premise that not
more than two per cent of persons wounded in the liver
recover, or from " not less than a certain proportion of
the JTs are P's," or from " no very large nor very
small proportion, etc." In short, p is subject to every
kind of indeterminacy; it simply excludes some ratios
and admits the possibility of the rest.

The analogy between syllogism and what is here called
probable deduction is certainly genuine and important ;
yet how wide the differences between the two modes of
inference are, will appear from the following considera
tions : —

1. The logic of probability is related to ordinary syllo
gistic as the quantitative to the qualitative branch of the
same science. Necessary syllogism recognizes only the
inclusion or non-inclusion of one class under another;
but probable inference takes account of the proportion

128 A THEORY OF PROBABLE INFERENCE.

of one class which is contained under a second. It is
like the distinction between protective geometry, which
asks whether points coincide or not, and metric geome
try, which determines their distances.

2. For the existence of ordinary syllogism, all that is
requisite is that we should be able to say, in some sense,
that one term is contained in another, or that one object
stands to a second in one of those relations : " better
than," " equivalent to," etc., which are termed transitive
because if A is in any such relation to B, and B is in
the same relation to (7, then A is in that relation to 0.
The universe might be all so fluid and variable that
nothing should preserve its individual identity, and that
no measurement should be conceivable ; and still one
portion might remain inclosed within a second, itself
inclosed within a third, so that a syllogism would be
possible. But probable inference could not be made in
such a universe, because no signification would attach to
the words " quantitative ratio." For that there must be
counting ; and consequently units must exist, preserving
their identity and variously grouped together.

3. A cardinal distinction between the two kinds of
inference is, that in demonstrative reasoning the con
clusion follows from the existence of the objective facts
laid down in the premises ; while in probable reasoning
these facts in themselves do not even render the con
clusion probable, but account has to be taken of various
subjective circumstances, — of the manner in which the
premises have been obtained, of there being no counter
vailing considerations, etc. ; in short, good faith and hon
esty are essential to good logic in probable reasoning.

When the partial rule that the proposition p of the
M9s are P's is applied to show with probability p that
8 is a P, it is requisite, not merely that S should le an

A THEORY OF PROBABLE INFERENCE. 129

Jlf, but also that it should be an instance drawn at ran
dom from among the M'a. Thus, there being four aces
in a picquet pack of thirty-two cards, the chance is one
eighth that a given card not looked at is an ace ; but
this is only on the supposition that the card has been
drawn at random from the whole pack. If, for instance,
it had been drawn from the cards discarded by the
players at piquet or euchre, the probability would be
quite different. The instance must be drawn at ran
dom. Here is a maxim of conduct. The volition of
the reasoner (using what machinery it may) has to
choose S so that it shall be an M\ but he ought to
restrain himself from all further preference, and not
allow his will to act in any way that might tend to
settle what particular M is taken, but should leave that
to the operation of chance. Willing and wishing, like
other operations of the mind, are general and imperfectly
determinate. I wish for a horse, — for some particular
kind of horse perhaps, but not usually for any individual
one. I will to act in a way of which I have a general
conception ; but so long as my action conforms to that
general description, how it is further determined I do
not care. Now in choosing the instance 8, the general
intention (including the whole plan of action) should
be to select an M, but beyond that there should be no
preference ; and the act of choice should be such that if
it were repeated many enough times with the same in
tention, the result would be that among the totality of
selections the different sorts of M's would occur with
the same relative frequencies as in experiences in which
volition does not intermeddle at all. In cases in which
it is found difficult thus to restrain the will by a direct
effort, the apparatus of games of chance, — a lottery-
wheel, a roulette, cards, or dice, — may be called to our

130 A THEORY OF PROBABLE INFERENCE.

aid. Usually, however, in making a simple probable
deduction, we take that instance in which we happen at
the time to be interested. In such a case, it is our
interest that fulfils the function of an apparatus for
random selection ; and no better need be desired, so
long as we have reason to deem the premise " the pro
portion p of the M's are P's " to be equally true in
regard to that part of the M's which are alone likely
ever to excite our interest.

Nor is it a matter of indifference in what manner the
other premise has been obtained. A card being drawn
at random from a picquet pack, the chance is one-eighth
that it is an ace, if we have no other knowledge of it.
But after we have looked at the card, we can no longer
reason in that way. That the conclusion must be drawn
in advance of any other knowledge on the subject is
a rule that, however elementary, will be found in the
sequel to have great importance.

4. The conclusions of the two modes of inference like
wise differ. One is necessary ; the other only probable.
Locke, in the "Essay concerning Human Understanding,"
hints at the correct analysis of the nature of probability.
After remarking that the mathematician positively knows
that the sum of the three angles of a triangle is equal to
two right angles because he apprehends the geometrical
proof, he then continues : " But another man who never
took the pains to observe the demonstration, hearing a
mathematician, a man of credit, affirm the three angles
of a triangle to be equal to two right ones, assents to it ;
that is, receives it for true. In which case, the founda
tion of his assent is the probability of the thing, the proof
being such as, for the most part, carries truth with it ;
the man on whose testimony he receives it not being wont
to affirm anything contrary to or besides his knowledge,

A THEOEY OF PROBABLE INFERENCE. 131

especially in matters of this kind." Those who know
Locke are accustomed to look for more meaning in his
words than appears at first glance. There is an allusion
in this passage to the fact that a probable argument is
always regarded as belonging to a genus of arguments.
Tliis is, in fact, true of any kind of argument. For the
belief expressed by the conclusion is determined or caused
by the belief expressed by the premises. There is, there
fore, some general rule according to which the one suc
ceeds the other. But, further, the reasoner is conscious
of there being such a rule, for otherwise he would not
know he was reasoning, and could exercise no attention
or control ; and to such an involuntary operation the
name reasoning is very properly not applied. In all
cases, then, we are conscious that our inference belongs
to a general class of logical forms, although we are not
necessarily able to describe the general class. The dif
ference between necessary and probable reasoning is that
in the one case we conceive that such facts as are ex
pressed by the premises are never, in the whole range of
possibility, true, without another fact, related to them as
our conclusion is to our premises, being true likewise ;
while in the other case we merely conceive that, in rea
soning as we do, we are following a general maxim that
will usually lead us to the truth.

So long as there are exceptions to the rule that all
men wounded in the liver die, it docs not necessarily
follow that because a given man is wounded in the liver
he cannot recover. Still, we know that if we were to
reason in that way, we should be following a mode of
inference which would only lead us wrong, in the long
run, once in fifty times ; and this is what we mean when
we say that the probability is one out of fifty that the
man will recover. To say, then, that a proposition has

132 A THEORY OF PROBABLE INFERENCE.

the probability p means that to infer it to be true would
be to follow an argument such as would carry truth with
it in the ratio of frequency p.

It is plainly useful that we should have a stronger
feeling of confidence about a sort of inference which will
oftener lead us to the truth than about an inference that
will less often prove right, — and such a sensation we do
have. The celebrated law of Fechner is, that as the
force acting upon an organ of sense increases in geo
metrical progression, the intensity of the sensation in
creases in arithmetical progression. In this case the
odds (that is, the ratio of the chances in favor of a
conclusion to the chances against it) take the place of
the exciting cause, while the sensation itself is the feel
ing of confidence. When two arguments tend to the
same conclusion, our confidence in the latter is equal to
the sum of what the two arguments separately would
produce ; the odds are the product of the odds in favor
of the two arguments separately. When the value of the
odds reduces to unity, our confidence is null ; when the
odds are less than unity, we have more or less confidence
in the negative of the conclusion.

II.

The principle of probable deduction still applies when
S, instead of being a single Jtf, is a set of M' s, — n
in number. The reasoning then takes the following
form : —

FORM III.
Complex Probable Deduction.

Among all sets of n M's, the proportion q consist each of
m P's and of n — m not-P's ;

A THEOEY OF PROBABLE INFERENCE. 133

S, Sr, Sff, etc. form a set of n objects drawn at random
from among the M*s :

Hence, the probability is q that among S, S', S", etc. there
are m P's and n — m not-P's.

In saying that S, S', S", etc. form a set drawn at ran
dom, we here mean that not only are the different in
dividuals drawn at random, but also that they are so
drawn that the qualities which may belong to one have
no influence upon the selection of any other. In other
words, the individual drawings are independent, and the
set as a whole is taken at random from among all possi
ble sets of n M's. In strictness, this supposes that the
same individual may be drawn several times in the same
set, although if the number of M's is large compared
with n, it makes no appreciable difference whether this
is the case or not.

The following formula expresses the proportion, among
all sets of n M's, of those which consist of m P's and
n — m not-P's. The letter r denotes the proportion of
P's among the M's, and the sign of admiration is used
to express the continued product of all integer numbers
from 1 to the number after which it is placed. Thus,
4 ! = 1 . 2 . 3 . 4 = 24, etc. The formula is

r"» (1 — r)n-m

q = nl X — : X —

m I (n — m) I

As an example, let us assume the proportion r = f
and the number of M's in a set n = 15. Then the
values of the probability q for different numbers, m, of
P's, are fractions having for their common denominator
14,348,907, and for their numerators as follows : —

134 A THE GET OF PEOBABLE INFERENCE.

Numerator of q.

420

3640

21840

96096

320320

823680

1\ umerator of q.

1667360

2562560

3075072

2795520

1863680

860160

122880

32768

A very little mathematics would suffice to show that,
r and n being fixed, q always reaches its maximum value
with that value of m that is next less than (n + 1)?*,* and
that q is very small unless m has nearly this value.

Upon these facts is based another form of inference to
which I give the name of statistical deduction. Its gen
eral formula is as follows : —

FORM IV.

{Statistical Deduction.
The proportion r of the M ?s are P?s ;

Sf, £;/, S>", etc., are a numerous set, taken at random
from among the Hf's :

Hence, probably and approximately, the proportion r of
the S's are P's.

As an example, take this : —

A little more than half of all human births are males ;
Hence, probably a little over half of all the births in New
York during any one year are males.

We have now no longer to deal with a mere probable
inference, but with a probable approximate inference.

* In case (n-f- l)ris a whole number, q has equal valut-s for m =
(TI -|- 1) r and for m — (n -f- 1) r — 1.

A THEORY OF PROBABLE INFERENCE. 135

This conception is a somewhat complicated one, meaning
that the probability is greater according as the limits of
approximation are wider, conformably to the mathemati
cal expression for the values of q.

This conclusion has no meaning at all unless there be
more than one instance ; and it has hardly any meaning
unless the instances are somewhat numerous. When
this is the case, there is a more convenient way of ob
taining (not exactly, but quite near enough for all practi
cal purposes) either a single value of q or the sum of
successive values from m = m\ to m = m2 inclusive. The
rule is first to calculate two quantities which may con
veniently be called ^ and t.2 according to these form
ula : —

^ ffll _ (n _j_ 1) r
tl = y2nr(l— rj

1 -f m2 — (n + 1) r

where w2 > Wi- Either or both the quantities ^ and ^
may be negative. Next with each of these quantities
enter the table below, and take out | 9^ and | 9t2 and
give each the same sign as the t from which it is derived

Then

2 q = l ®t2 — 1 ©*r

136 A THEORY OF PROBABLE INFERENCE.

TaUeofSt =

—p

dt.

0.0

0.000

0.1

0.112

0.2

0.223

0.3

0.329

0.4

0.428

0.5

0.520

0.6

0.604

0.7

0.678

0.8

0.742

0.9

0.797

1.0

0.843

1.0

0.843

1.1

0.880

1.2

0.910

1.3

0.934

1.4

0.952

1.5

0.966

1.6

0.976

1.7

0.984

1.8

0.989

1.9

0.993

2.0

0.995

2.0

0.99532

2.1

0.99702

2.2

0.99814

2.3

0.9988"6

2.4

0.99931

2.5

0.99959

2.6

0.99976

2.7

0.99987

2.8

0.99992

2.9

0.99996

3.0

0.99998

0.999999989

0.9999999999984

0.999999999999999982

0.999999999999999999999958

The principle of statistical deduction is that these two
proportions, — namely, that of the P's among the if's,
and that of the P's among the £'s, — are probably and
approximately equal. If, then, this principle justifies our
inferring the value of the second proportion from the
known value of the first, it equally justifies our inferring
the value of the first from that of the second, if the first

A THEORY OF PROBABLE INFERENCE. 137

is unknown but the second has been observed. We
thus obtain the following form of inference : —

FORM V.
Induction.

S1, Sfl, S'", etc., form a numerous set taken at random
from among the M's ;

S1, S", S'", etc., are found to be — the proportion p of
them — P's :

Hence, probably and approximately the same proportion, p,
of the M'a are P's.

The following are examples. From a bag of coffee a
handful is taken out, and found to have nine tenths of
the beans perfect ; whence it is inferred that about nine-
tenths of all the beans in the bag are probably perfect.
The United States Census of 1870 shows that of native
white children under one year old, there were 478,774
males to 463,320 females ; while of colored children of
the same age there were 75,985 males to 76,637 females.
We infer that generally there is a larger proportion of
female births among negroes than among whites.

When the ratio p is unity or zero, the inference is an
ordinary induction ; and 1 ask leave to extend the term
induction to all such inference, whatever be the value of
p. It is, in fact, inferring from a sample to the whole
lot sampled. These two forms of inference, statistical
deduction and induction, plainly depend upon the same
principle of equality of ratios, so that their validity is the
same. Yet the nature of the probability in the two cases
is very different. In the statistical deduction, we know
that among the whole body of M 's the proportion of P's
is p ; we say, then, that the S9s being random drawings

138 A THEORY OF PROBABLE INFERENCE.

of Jf'sare probably P's in about the same proportion,
— and though this may happen not to be so, yet at any
rate, on continuing the drawing sufficiently, our pre
diction of the ratio will be vindicated at last. On the
other hand, in induction we say that the proportion p of
the sample being P's, probably there is about the same
proportion in the whole lot ; or at least, if this happens
not to be so, then on continuing the drawings the in
ference will be, not vindicated as in the other case, but
modified so as to become true. The deduction, then,
is probable in this sense, that though its conclusion may
in a particular case be falsified, yet similar conclusions
(with the same ratio p) would generally prove approxi
mately true; while the induction is probable in this
sense, that though it may happen to give a false con
clusion, yet in most cases in which the same precept of
inference was followed, a different and approximately
true inference (with the right value of p) would be
drawn.

IY.

Before going any further with the study of Form V.,
I wish to join to it another extremely analogous form.

We often speak of one thing being very much like
another, and thus apply a vague quantity to resemblance.
Even if qualities are not subject to exact numeration,
we may conceive them to be approximately measurable.
We may then measure resemblance by a scale of num
bers from zero up to unity. To say that S has a
1-likeness to a P will mean that it has every character
of a P, and consequently is a P. To say that it has a
0-likeness will imply total dissimilarity. We shall then
be able to reason as follows : —

A THEORY OF PROBABLE INFERENCE. 139

FORM II. (bis).
Simple probable deduction in depth.

Every M has the simple mark P ;
The >S"s have an r-likeness to the M's :
Hence, the probability is r that every S is P.

It would be difficult, perhaps impossible, to adduce an
example of such kind of inference, for the reason that
simple marks are not known to us. We may, however,
illustrate the complex probable deduction in depth (the
general form of which it is not worth while to set down)
as follows : I forget whether, in the ritualistic churches,
a bell is tinkled at the elevation of the Host or not.
Knowing, however, that the services resemble somewhat
decidedly those of the Roman Mass, I think that it is not
unlikely that the bell is used in the ritualistic, as in the
Roman, churches.

We shall also have the following : —

FORM IV. (bis).
Statistical deduction in depth.

Every M has, for example, the numerous marks P7, Prr,
P", etc.

S has an r-likeness to the M's :

Hence, probably and approximately, S has the proportion r
of the marks P', P", P'", etc.

For example, we know that the French and Italians
are a good deal alike in their ideas, characters, tempera
ments, genius, customs, institutions, etc., while they also
differ very markedly in all these respects. Suppose, then,
that I know a boy who is going to make a short trip
through France and Italy ; I can safely predict that
among the really numerous though relatively few res-

140 A THEORY OF PROBABLE INFERENCE.

pects in which he will be able to compare the two people,
about the same degree of resemblance will be found.

Both these modes of inference are clearly deductive.
When r = 1, they reduce to Barbara.1

Corresponding to induction, we have the following
mode of inference: —

FORM V. (bis).
Hypothesis.

M has, for example, the numerous marks P', P", P/;/, etc.
S has the proportion r of the marks P', P", P"!, etc. :
Hence, probably and approximately, £has an r-likeness to M.

Thus, we know, that the ancient Mound-builders of
North America present, in all those respects in which we
have been able to make the comparison, a limited degree
of resemblance with the Pueblo Indians. The inference
is, then, that in all respects there is about the same de
gree of resemblance between these races.

If I am permitted the extended sense which I have
given to the word " induction," this argument is simply
an induction respecting qualities instead of respecting

1 When r = 0, the last form becomes

M has all the marks P ;

S has no mark of M :

Hence, S has none of the marks P.

"When the universe of marks is unlimited (see a note appended to this
paper for an explanation of this expression), the only way in which two
terms can fail to have a common mark is by their together filling the uni
verse of things ; and consequently this form then becomes,

3/isP;

Every non-£ is M:

Hence, every non-S is P.

This is one of De Moi-gan's syllogisms.

In putting r = 0 in Form II. (bis) it must be noted that, since P is
simple in depth, to say that S is not P is to say that it has no mark of P.

A THEORY OF PROBABLE INFERENCE. 141

things. In point of fact P', P", P"', etc. constitute a
random sample of the characters of M, and the ratio r
of them being found to belong to $ the same ratio of all
the characters of M are concluded to belong to S. This
kind of argument, however, as it actually occurs, differs
very much from induction, owing to the impossibility
of simply counting qualities as individual things are
counted. Characters have to be weighed rather than
counted. Thus, antimony is bluish-gray : that is a char
acter. Bismuth is a sort of rose-gray; it is decidedly
different from antimony in color, and yet not so very
different as gold, silver, copper, and tin are.

I call this induction of characters hypothetic inference,
or, briefly, hypothesis. This is perhaps not a very happy
designation, yet it is difficult to find a better. The term
"hypothesis" has many well established and distinct
meanings. Among these is that of a proposition believed
in because its consequences agree with experience. This
is the sense in which Newton used the word when he
said, Hypotheses non jingo. He meant that he was merely
giving a general formula for the motions of the heavenly
bodies, but was not undertaking to mount to the causes
of the acceleration they exhibit. The inferences of
Kepler, on the other hand, were hypotheses in this sense;
for he traced out the miscellaneous consequences of the
supposition that Mars moved in an ellipse, with the sun
at the focus, and showed that both the longitudes and the
latitudes resulting from this theory were such as agreed
with observation. These two components of the motion
were observed ; the third, that of approach to or regression
from the earth, was supposed. Now, if in Form V. (bis)
we put r = 1, the inference is the drawing of a hypothesis
in this sense. I take the liberty of extending the use of
the word by permitting r to have any value from zero to

142 A THEORY OF PKOBABLE INFERENCE.

unity. The term is certainly not all that could be de
sired ; for the word hypothesis, as ordinarily used, carries
with it a suggestion of uncertainty, and of something to
be superseded, which does not belong at all to my use of
it. But we must use existing language as best we may,
balancing the reasons for and against any mode of ex
pression, for none is perfect ; at least the term is not
so utterly misleading as " analogy " would be, and with
proper explanation it will, I hope, be understood.

The following examples will illustrate the distinction
between statistical deduction, induction, and hypothesis.
If I wished to order a font of type expressly for the
printing of this book, knowing, as I do, that in all Eng
lish writing the letter e occurs oftener than any other
letter, I should want more e's in my font than other
letters. For what is true of all other English writing is
no doubt true of these papers. This is a statistical de
duction. But then the words used in logical writings are
rather peculiar, and a good deal of use is made of single
letters. I might, then, count the number of occurrences
of the different letters upon a dozen or so pages of the
manuscript, and thence conclude the relative amounts of
the different kinds of type required in the font. That
would be inductive inference. If now I were to order
the font, and if, after some days, I were to receive a box
containing a large number of little paper parcels of very
different sizes, I should naturally infer that this was the
font of types I had ordered ; and this would be hypothetic
inference. Again, if a dispatch in cipher is captured, and
it is found to be written with twenty-six characters, one
of which occurs much more frequently than any of the

A THEORY OF PROBABLE INFERENCE. 143

others, we are at once led to suppose that each charac
ter represents a letter, and that the one occurring so fre
quently stands fer e. This is also hypothetic inference.

We are thus led to divide all probable reasoning into
deductive and ampliative, and further to divide ampliative
reasoning into induction and hypothesis. In deductive
reasoning, though the predicted ratio may be wrong in a
limited number of drawings, yet it will be approximately
verified in a larger number. In ampliative reasoning the
ratio may be wrong, because the inference is based on but
a limited number of instances ; but on enlarging the
sample the ratio will be changed till it becomes approxi
mately correct. In induction, the instances drawn at
random are numerable things ; in hypothesis they are
characters, which are not capable of strict enumeration,
but have to be otherwise estimated.

This classification of probable inference is connected
with a preference for the copula of inclusion over those
used by Miss Ladd and by Mr. Mitchell.1 De Morgan
established eight forms of simple propositions ; and from
a purely formal point of view no one of these has a right
to be considered as more fundamental than any other.
But formal logic must not be too purely formal ; it must
represent a fact of psychology, or else it is in danger of
degenerating into a mathematical recreation. The cate
gorical proposition, "every man is mortal," is but a modifi
cation of the hypothetical proposition, " if humanity, then
mortality ;" and since the very first conception from which
logic springs is that one proposition follows from another,
I hold that "if A, then B" should be taken as the typical
form of judgment. Time flows ; and, in time, from one
state of belief (represented by the premises of an argu-

1 I do not here speak of Mr. Jeyons, because my objection to the copula
of identity is of a somewhat different kind.

144 A THEOKY OF PROBABLE INFERENCE.

ment) another (represented by its conclusion) is de
veloped. Logic arises from this circumstance, without
which we could not learn anything nor correct any
opinion. To say that an inference is correct is to say
that if the premises are true the conclusion is also true ;
or that every possible state of things in which the prem
ises should be true would be included among the possible
states of things in which the conclusion would be true.
We are thus led to the copula of inclusion. But the
main characteristic of the relation of inclusion is that it
is transitive, — that is, that what is included in some
thing included in anything is itself included in that
thing ; or, that if A is B and B is (7, then A is 0. We
thus get Barbara as the primitive type of inference.
Now in Barbara we have a Rule, a Case under the Rule,
and the inference of the Result of that rule in that case.
For example : —

Rule. All men are mortal ;
Case. Enoch was a man.
Result. Enoch was mortal.

The cognition of a rule is not necessarily conscious,
but is of the nature of a habit, acquired or congenital.
The cognition of a case is of the general nature of a
sensation; that is to say, it is something which comes
up into present consciousness. The cognition of a result
is of the nature of a decision to act in a particular way
on a given occasion.1 In point of fact, a syllogism, in
Barbara virtually takes place when we irritate the foot
of a decapitated frog. The connection between the af
ferent and efferent nerve, whatever it may be, constitutes
a nervous habit, a rule of action, which is the physio-

1 See my paper on " How to make our ideas clear." — Popular Science
Monthly, January, 1878.

A THEORY OF PROBABLE INFERENCE. 145

logical analogue of the major premise. The disturbance
of the ganglionic equilibrium, owing to the irritation, is
the physiological form of that which, psychologically con
sidered, is a sensation ; and, logically considered, is the
occurrence of a case. The explosion through the efferent
nerve is the physiological form of that which psychologi
cally is a volition, and logically the inference of a result.
When we pass from the lowest to the highest forms of
inervation, the physiological equivalents escape our ob
servation ; but, psychologically, we still have, first, habit,
— which in its highest form is understanding, and which
corresponds to the major premise of Barbara; we have,
second, feeling, or present consciousness, corresponding
to the minor premise of Barbara; and we have, third,
volition, corresponding to the conclusion of the same
mode of syllogism. Although these analogies, like all
very broad generalizations, may seem very fanciful at
first sight, yet the more the reader reflects upon them
the more profoundly true I am confident they will appear.
They give a significance to the ancient system of formal
logic which no other can at all share.

Deduction proceeds from Rule and Case to Result ; it
is the formula of Volition. Induction proceeds from Case
and Result to Rule ; it is the formula of the formation of
a habit or general conception, — a process which, psycho
logically as well as logically, depends on the repetition of
instances or sensations. Hypothesis proceeds from Rule
and Result to Case ; it is the formula of the acquirement
of secondary sensation, — a process by which a confused
concatenation of predicates is brought into order under
a synthetizing predicate.

We usually conceive Nature to be perpetually making
deductions in Barbara. This is our natural and anthro
pomorphic metaphysics. We conceive that there are

146 A THEOKY OF PROBABLE INFERENCE.

Laws of Nature, which are her Rules or major premises.
We conceive that Cases arise under these laws ; these
cases consist in the predication, or occurrence, of causes,
which are the middle terms of the syllogisms. And,
finally, we conceive that the occurrence of these causes,
by virtue of the laws of Nature, result in effects which
are the conclusions of the syllogisms. Conceiving of
nature in this way, we naturally conceive of science as
having three tasks, — (1) the discovery of Laws, which
is accomplished by induction ; (2) the discovery of Causes,
which is accomplished by hypothetic inference ; and (3)
the prediction of Effects, which is accomplished by de
duction. It appears to me to be highly useful to select
a system of logic which shall preserve all these natural
conceptions.

It may be added that, generally speaking, the conclu
sions of Hypothetic Inference cannot be arrived at in
ductively, because their truth is not susceptible of direct
observation in single cases. Nor can the conclusions of
Inductions, on account of their generality, be reached by
hypothetic inference. For instance, any historical fact,
as that Napoleon Bonaparte once lived, is a hypothesis ;
we believe the fact, because its effects — I mean current
tradition, the histories, the monuments, etc. — are ob
served. But no mere generalization of observed facts
could ever teach us that Napoleon lived. So we induc
tively infer that every particle of matter gravitates toward
every other. Hypothesis might lead to this result for
any given pair of particles, but it never could show that
the law was universal.

VI.

We now come to the consideration of the Rules which
have to be followed in order to make valid and strong

A THEOKY OF PKOBABLE INFERENCE. 147

Inductions and Hypotheses. These rules can all be re
duced to a single one ; namely, that the statistical deduc
tion of which the Induction or Hypothesis is the inversion,
must be valid and strong.

We have seen that Inductions and Hypotheses are in
ferences from the conclusion and one premise of a sta
tistical syllogism to the other premise. In the case of
hypothesis, this syllogism is called the explanation. Thus
in one of the examples used above, we suppose the cryp
tograph to be an English cipher, because, as we say, this
explains the observed phenomena that there are about
two dozen characters, that one occurs more frequently
than the rest, especially at the ends of words, etc. The
explanation is, —

Simple English ciphers have certain peculiarities ;

This is a simple English cipher :

Hence, this necessarily has these peculiarities.

This explanation is present to the mind of the reasoner,
too ; so much so, that we commonly say that the hypo
thesis is adopted for the sake of the explanation. Of
induction we do not, in ordinary language, say that it
explains phenomena; still, the statistical deduction, of
which it is the inversion, plays, in a general way, the
same part as the explanation in hypothesis. From a
barrel of apples, that I am thinking of buying, I draw
out three or four as a sample. If I find the sample some
what decayed, I ask myself, in ordinary language, not
' Why is this ? " but « How is this ? » And I answer
that it probably comes from nearly all the apples in the
barrel being in bad condition. The distinction between
the « Why" of hypothesis and the " How" of induction
is not very great ; both ask for a statistical syllogism, of
which the observed fact shall be the conclusion, the

148 A THEOEY OF PROBABLE INFERENCE.

known conditions of the observation one premise, and
the inductive or hypothetic inference the other. This
statistical syllogism may be conveniently termed the ex
planatory syllogism.

In order that an induction or hypothesis should have
any validity at all, it is requisite that the explanatory
syllogism should be a valid statistical deduction. Its
conclusion must not merely follow from the premises,
but follow from them upon the principle of probability.
The inversion of ordinary syllogism does not give rise
to an induction or hypothesis. The statistical syllogism
of Form 1Y. is invertlble, because it proceeds upon the
principle of an approximate equality between the ratio
of P's in the whole class and the ratio in a well-drawn
sample, and because equality is a convertible relation.
But ordinary syllogism is based upon the property of the
relation of containing and contained, and that is not a
convertible relation. There is, however, a way in which
ordinary syllogism may be inverted ; namely, the con
clusion and either of the premises may be interchanged
by negativing each of them. This is the way in which
the indirect, or apagogical,1 figures of syllogism are de
rived from the first, and in which the modus tollens is
derived from the modus ponens. The following schemes

show this : —

First Figure.

Rule. AllJfisP;
Case. S is M :
Result. S is P.

Second Figure.
Rule. AllJfisP;

Denial of Result. S is not P :
Denial of Case. S is not M.

Third Figure.

Denial of Result. S is not P ;
Case. /Sis M:

Denial of Rule. Some M is

not P.

1 From apagoge, Aristotle's name for the rcductio ad alsurdum.

A THEORY OF PROBABLE INFERENCE. 149

Modus Ponens.

Rule. If A is true, C is true ;
Case. In a certain case A is true :
Result. .'. In that case C is true.

Modus Tollens.

Rule. If A is true, C is

true;
Denial of Result. In a certain

case C is not true :
Denial of Case. .'. In that

case A is not true.

Modus Innominatus.

Case. In a certain case A is

true;
Denial of Result. In that case

C is not true :
Denial of Rule. .'. If A is true,

C is not necessarily true.

Now suppose we ask ourselves what would be the re
sult of thus apagogically inverting a statistical deduction.
Let us take, for example, Form IV : —

The $'s are a numerous random sample of the M 's ;

The proportion r of the M's are P's :

Hence, probably about the proportion r of the S's are P's.

The ratio r, as we have already noticed, is not neces
sarily perfectly definite ; it may be only known to have
a certain maximum or minimum ; in fact, it may have
any kind of indeterminacy. Of all possible values be
tween 0 and 1, it admits of some and excludes others.
The logical negative of the ratio r is, therefore, itself a
ratio, which we may name p ; it admits of every value
which r excludes, and excludes every value of which r
admits. Transposing, then, the major premise and con
clusion of our statistical deduction, and at the same time
denying both, we obtain the following inverted form : —

150 A THEORY OF PROBABLE INFERENCE.

The S's are a numerous random sample of the M 'a ;

The proportion p of the S's are .P's :

Hence, probably about the proportion p of the M 'a are P's.1

But this coincides with the formula of Induction.
Again, let us apagogically invert the statistical deduction
of Form IV. (fo's). This form is, —

Every M has, for example, the numerous marks P;, P",
P"', etc.

S has an r-likeness to the M'a :

Hence, probably and approximately, S has the proportion
r of the marks Pf, P", P"1, etc.

Transposing the minor premise and conclusion, at the
same time denying both, we get the inverted form, —

Every M has, for example, the numerous marks Pf, Prf,
P"', etc.

S has the proportion p of the marks P', P", Pf!f, etc. :
Hence, probably and approximately, S has a p-likeness to

the class of M ?s.

This coincides with the formula of Hypothesis. Thus
we see that Induction and Hypothesis are nothing but
the apagogical inversions of statistical deductions. Ac
cordingly, when r is taken as 1, so that p is "less than 1,"
or when r is taken as 0, so that p is " more than 0," the
induction degenerates into a syllogism of the third figure
and the hypothesis into a syllogism of the second figure.

1 The conclusion of the statistical deduction is here regarded as being
"the proportion r of the S's are P's," and the words "probably about"
as indicating the modality with which this conclusion is drawn and held
for true. It would be equally true to consider the "probably about" as
forming part of the contents of the conclusion ; only from that point of
view the inference ceases to be probable, and becomes rigidly necessary,
and its apagogical inversion is also a necessary inference presenting no
particular interest.

A THEORY OF PEOBABLE INFERENCE. 151

In these special cases, there is no very essential difference
between the mode of reasoning in the direct and in the
apagogical form. But, in general, while the probability
of the two forms is precisely the same, — in this sense,
that for any fixed proportion of _P's among the M'a
(or of marks of jS9s among the marks of the M's) the
probability of any given error in the concluded value is
precisely the same in the indirect as it is in the direct
form, — yet there is this striking difference, that a multi
plication of instances will in the one case confirm, and
in the other modify, the concluded value of the ratio.

We are thus led to another form for our rule of validity
of ampliative inference ; namely, instead of saying that
the explanatory syllogism must be a good probable de
duction, we may say that the syllogism of which the
induction or hypothesis is the apagogical modification
(in the traditional language of logic, the reduction) must
be valid.

Probable inferences, though valid, may still differ in
their strength. A probable deduction has ,a greater or
less probable error in the concluded ratio. When r is a
definite number the probable error is also definite ; but
as a general rule we can only assign maximum and mini
mum values of the probable error. The probable error
is, in fact, —

0.477 V^-^
n

where n is the number of independent instances. The
same formula gives the probable error of an induction or
hypothesis ; only that in these cases, r being wholly inde
terminate, the minimum value is zero, and the maximum
is obtained by putting r = J.

152 A THEORY OF PROBABLE INFERENCE.

VII.

Although the rule given above really contains all the
conditions to which Inductions and Hypotheses need to
conform, yet inasmuch as there are many delicate ques
tions in regard to the application of it, and particularly
since it is of that nature that a violation of it, if not
too gross, may not absolutely destroy the virtue of the
reasoning, a somewhat detailed study of its requirements
in regard to each of the premises of the argument is still
needed.

The first premise of a scientific inference is that certain
things (in the case of induction) or certain characters
(in the case of hypothesis) constitute a fairly chosen
sample of the class of things or the run of characters
from which they have been drawn.

The rule requires that the sample should be drawn at
random and independently from the whole lot sampled.
That is to say, the sample must be taken according to a
precept or method which, being applied over and over
again indefinitely, would in the long run result in the
drawing of any one set of instances as often as any other
set of the same number.

The needfulness of this rule is obvious ; the difficulty
is to know how we are to carry it out. The usual method
is mentally to run over the lot of objects or characters to
be sampled, abstracting our attention from their peculi
arities, and arresting ourselves at this one or that one
from motives wholly unconnected with those peculiarities.
But this abstention from a further determination of our
choice often demands an effort of the will that is beyond
our strength ; and in that case a mechanical contrivance
may be called to our aid. We may, for example, number
all the objects of the lot, and then draw numbers by

A THEORY OF PROBABLE INFERENCE. 153

means of a roulette, or other such instrument. We may
even go so far as to say that this method is the type of
all random drawing ; for when we abstract our attention
from the peculiarities of objects, the psychologists tell us
that what we do is to substitute for the images of sense
certain mental signs, and when we proceed to a random
and arbitrary choice among these abstract objects we are
governed by fortuitous determinations of the nervous sys
tem, which in this case serves the purpose of a roulette.

The drawing of objects at random is an act in which
honesty is called for ; and it is often hard enough to be
sure that we have dealt honestly with ourselves in the
matter, and still more hard to be satisfied of the honesty
of another. Accordingly, one method of sampling has
come to be preferred in argumentation ; namely, to take
of the class to be sampled all the objects of which we
have a sufficient knowledge. Sampling is, however, a
real art, well deserving an extended study by itself : to
enlarge upon it here would lead us aside from our main
purpose.

Let us rather ask what will be the effect upon inductive
inference of an imperfection in the strictly random char
acter of the sampling. Suppose that, instead of using
such a precept of selection that any one M would in the
long run be chosen as often as any other, we used a
precept which would give a preference to a certain half
of the J/'s, so that they would be drawn twice as often
as the rest. If we were to draw a numerous sample by
such a precept, and if we were to find that the proportion
p of the sample consisted of JP's, the inference that we
should be regularly entitled to make would be, that among
all the M'a, counting the preferred half for two each, the
proportion p would be P's. But this regular inductive
inference being granted, from it we could deduce by

154 A THEORY OF PROBABLE INFERENCE.

arithmetic the further conclusion that, counting the M's
for one each, the proportion of P's among them must
(p being over f ) lie between | p + \ and f p — J. Hence,
if more than two thirds of the instances drawn by the use
of the false precept were found to be P's, we should be
entitled to conclude that more than half of all the M's
were P's. Thus, without allowing ourselves to be led
away into a mathematical discussion, we can easily see
that, in general, an imperfection of that kind in the
random character of the sampling will only weaken the
inductive conclusion, and render the concluded ratio less
determinate, but will not necessarily destroy the force
of the argument completely. In particular, when p ap
proximates towards 1 or 0, the effect of the imperfect
sampling will be but slight.

Nor must we lose sight of the constant tendency of the
inductive process to correct itself. This is of its essence.
This is the marvel of it. The probability of its conclusion
only consists in the fact that if the true value of the ratio
sought has not been reached, an extension of the induc
tive process will lead to a closer approximation. Thus,
even though doubts may be entertained whether one se
lection of instances is a random one, yet a different se
lection, made by a different method, will be likely to vary
from the normal in a different way, and if the ratios
derived from such different selections are nearly equal,
they may be presumed to be near the truth. This con
sideration makes it extremely advantageous in all ampli-
ative reasoning to fortify one method of investigation by
another.1 Still we must not allow ourselves to trust so

1 This I conceive to be all the truth there is in the doctrine of Bacon
and Mill regarding different Methods of Experimental Inquiry. The main
proposition of Bacon and Mill's doctrine is, that in order to prove that all
M 'a are P's, we should not only take random instances of the M' s and

A THEORY OF PROBABLE INFERENCE. 155

much to this virtue of induction as to relax our efforts
towards making our drawings of instances as random
and independent as we can. For if we infer a ratio from
a number of different inductions, the magnitude of its
probable error will depend very much more on the worst
than on the best inductions used.

We have, thus far, supposed that although the selection
of instances is not exactly regular, yet the precept fol
lowed is such that every unit of the lot would eventually
get drawn. But very often it is impracticable so to draw
our instances, for the reason that a part of the lot to be
sampled is absolutely inaccessible to our powers of obser
vation. If we want to know whether it will be profit
able to open a mine, we sample the ore ; but in advance
of our mining operations, we can obtain only what ore
lies near the surface. Then, simple induction becomes
worthless, and another method must be resorted to. Sup
pose we wish to make an induction regarding a series
of events extending from the distant past to the distant
future ; only those events of the series which occur within
the period of time over which available history extends
can be taken as instances. Within this period we may
find that the events of the class in question present some
uniform character ; yet how do we know but this uni
formity was suddenly established a little while before the
history commenced, or will suddenly break up a little
while after it terminates ? Now, whether the uniformity

examine them to see that they are Ps, but we should also take instances
of not-P's and examine them to see that they are not-J/'s. This is an
excellent way of fortifying one induction by another, when it is applicable;
but it is entirely inapplicable when r has any other value than 1 or 0.
For, in general, there is no connection between the proportion of M' s that
are Ps and the proportion of non-P's that are non-l/'s. A very small
proportion of calves may be monstrosities, and yet a very large proportion
of monstrosities may be calves.

156 A THEORY OF PROBABLE INFERENCE.

observed consists (1) in a mere resemblance between all
the phenomena, or (2) in their consisting of a disorderly
mixture of two kinds in a certain constant proportion, or
(3) in the character of the events being a mathematical
function of the time of occurrence, — in any of these cases
we can make use of an apagoge from the following proba
ble deduction : —

Within the period of time M, a certain event P occurs ;
S is a period of time taken at random from M, and more
than half as long :

Hence, probably the event P will occur within the time S.

Inverting this deduction, we have the following ampli-
ative inference : —

S is a period of time taken at random from M9 and more

than half as long ;

The event P does not happen in the time S :

Hence, probably the event P does not happen in the

period M.

The probability of the conclusion consists in this, that
we here follow a precept of inference, which, if it is very
often applied, will more than half the time lead us right.
Analogous reasoning would obviously apply to any por
tion of an unidimensional continuum, which might be
similar to periods of time. This is a sort of logic which
is often applied by physicists in what is called extrapola
tion of an empirical law. As compared with a typical
induction, it is obviously an excessively weak kind of in
ference. Although indispensable in almost every branch
of science, it can lead to no solid conclusions in regard to
what is remote from the field of direct perception, unless
it be bolstered up in certain ways to which we shall have
occasion to refer further on.

A THEORY OF PROBABLE INFERENCE. 157

Let us now consider another class of difficulties in
regard to the rule that the samples must be drawn at
random and independently. In the first place, what if
the lot to be sampled be infinite in number ? In what
sense could a random sample be taken from a lot like
that ? A random sample is one taken according to a
method that would, in the long run, draw any one object
as often as any other. In what sense can such drawing
be made from an infinite class ? The answer is not far
to seek. Conceive a cardboard disk revolving in its own
plane about its centre, and pretty accurately balanced,
so that when put into rotation it shall be about1 as likely
to come to rest in any one position as in any other ; and
let a fixed pointer indicate a position on the disk: the
number of points on the circumference is infinite, and on
rotating the disk repeatedly the pointer enables us to
make a selection from this infinite number. Tbis means
merely that although the points are innumerable, yet
there is a certain order among them that enables us to
run them through and pick from them as from a very
numerous collection. In such a case, and in no other,
can an infinite lot be sampled. But it would be equally
true to say that a finite lot can be sampled only on
condition that it can be regarded as equivalent to an
infinite lot. For the random sampling of a finite class
supposes the possibility of drawing out an object, throw
ing it back, and continuing this process indefinitely ; so
that what is really sampled is not the finite collection of
things, but the unlimited number of possible drawings.

But though there is thus no insuperable difficulty in
sampling an infinite lot, yet it must be remembered that
the conclusion of inductive reasoning only consists in the

1 I say about, because the doctrine of probability only deals with ap
proximate evaluations.

158 A THEORY OF PROBABLE INFERENCE.

approximate evaluation of a ratio, so that it never can
authorize us to conclude that in an infinite lot sampled
there exists no single exception to a rule. Although all
the planets are found to gravitate toward one another,
this affords not the slightest direct reason for denying
that among the innumerable orbs of heaven there may
be some \vhich exert no such force. Although at no
point of space where we have yet been have we found
any possibility of motion in a fourth dimension, yet this
does not tend to show (by simple induction, at least)
that space has absolutely but three dimensions. Although
all the bodies we have had the opportunity of examining
appear to obey the law of inertia, this does not prove
that atoms and atomicules are subject to the same law.
Such conclusions must be reached, if at all, in some
other way than by simple induction. This latter may
show that it is unlikely that, in my lifetime or yours,
things so extraordinary should be found, but do not war
rant extending the prediction into the indefinite future.
And experience shows it is not safe to predict that such
and such a fact will never be met with.

If the different instances of the lot sampled are to
be drawn independently, as the rule requires, then the
fact that an instance has been drawn once must not
prevent its being drawn again. It is true that if the
objects remaining unchosen are very much more numer
ous than those selected, it makes practically no difference
whether they have a chance of being drawn again or not,
since that chance is in any case very small. Proba
bility is wholly an affair of approximate, not at all of
exact, measurement ; so that when the class sampled is
very large, there is no need of considering whether ob
jects can be drawn more than once or not. But in what
is known as " reasoning from analogy," the class sam-

A THEORY OF PROBABLE INFERENCE. 159

pled is small, and no instance is taken twice. For ex
ample : we know that of the major planets the Earth,
Mars, Jupiter, and Saturn revolve on their axes, and
we conclude that the remaining four, Mercury, Venus,
Uranus, and Neptune, probably do the like. This is
essentially different from an inference from what has
been found in drawings made hitherto, to what will be
found in indefinitely numerous drawings to be made
hereafter. Our premises here are that the Earth, Mars,
Jupiter, and Saturn are a random sample of a natural
class of major planets, — a class which, though (so far
as we know) it is very small, yet may be very extensive,
comprising whatever there may be that revolves in a
circular orbit around a great sun, is nearly spherical,
shines with reflected light, is very large, etc. Now the
examples of major planets that we can examine all ro
tate on their axes ; whence we suppose that Mercury,
Venus, Uranus, and Neptune, since they possess, so far
as we know, all the properties common to the natural
class to which the Earth, Mars, Jupiter, and Saturn be
long, possess this property likewise. The points to be
observed are, first, that any small class of things may be
regarded as a mere sample of an actual or possible large
class having the same properties and subject to the same
conditions; second, that while we do not know what all
these properties and conditions are, we do know some of
them, which some may be considered as a random sam
ple of all ; third, that a random selection without re
placement from a small class may be regarded as a true
random selection from that infinite class of which the
finite class is a random selection. The formula of the
analogical inference presents, therefore, three premises,
thus : —

160 A THEOEY OF PROBABLE INFERENCE.

/S7, S", S"f are a random sample of some undefined class X}
of whose characters Pf, P", P"1 are samples.
Q is P, P", P'".
S', S", S">, are 7?'s.
Hence, Q is an R.

"We have evidently here an induction and an hypothe
sis followed by a deduction ; thus, —

Every X is, for example, P',
P", P''7, etc.

Q is found to be P', P",
P>", etc.

Hence, hypothetically, Q is

*Sf/, £", /S^', etc., are samples

of the

S', S", S>», etc., are found
to be It's.

Hence, inductively, every X

is an R.
Hence, deductively, Q is an R.*'

An argument from analogy may be strengthened by
the addition of instance after instance to the premises,
until it loses its ampliative character by the exhaustion
of the class and becomes a mere deduction of that kind
called complete induction, in which, however, some shadow

* That this is really a correct analysis of the reasoning can be shown by
the theory of probabilities. For the expression

(P + g) ! (TT + P) • (P + ^)! (? + P)!

p \ q ! 7T ! p ! (p -j- TT -{- g -f- p) !

expresses at once the probability of two events ; namely, it expresses
first the probability that of p -f <7 objects drawn without replacement
from a lot consisting of p -f TT objects having the character E together
with q -f- p not having this character, the number of those drawn having
this character will bejo; and second, the same expression denotes the
probability that if among p -f- TT -f- q -f- p objects drawn at random from
an infmita class (containing no matter what proportion of It's to nori-72's),
it happens that p -f- TT have the character 72, then among any ^ -f- g of
them, designated at random, p will have the same character. Thus we
see that the chances in reference to drawing without replacement from a
finite class are precisely the same as those in reference to a class which
has been drawn at random from an infinite class.

A THEOKY OF PROBABLE INFEKENCE. 161

of the inductive character remains, as this name im
plies.

VIII.

Take any human being, at random, — say Queen Eliz
abeth. Now a little more than half of all the human
beings who have ever existed have been males ; but it
does not follow that it is a little more likely than not
that Queen Elizabeth was a male, since we know she was
a woman. Nor, if we had selected Julius Caesar, would
it be only a little more likely than not that he was a
male. It is true that if we were to go on drawing at
random an indefinite number of instances of human be
ings, a slight excess over one-half would be males. But
that which constitutes the probability of an inference is
the proportion of true conclusions among all those which
could be derived from the same precept. Now a precept
of inference, being a rule which the mind is to follow,
changes its character and becomes different when the
case presented to the mind is essentially different. When,
knowing that the proportion r of all M* s are P's, I draw
an instance, S, of an M, without any other knowledge of
whether it is a P or not, and infer with probability, r,
that it is P, the case presented to my mind is very
different from what it is if I have such other knowledge.
In short, I cannot make a valid probable inference with
out taking into account whatever knowledge I have (or,
at least, whatever occurs to my mind) that bears upon
the question.

The same principle may be applied to the statistical
deduction of Form IV. If the major premise, that the
proportion r of the 3/'s are P's, be laid down first,
before the instances of Ms are drawn, we really draw our
inference concerning those instances (that the proper-

162 A THEORY OF PROBABLE INFERENCE.

tion r of them will be P's) in advance of the drawing,
and therefore before we know whether they are P's or
not. But if we draw the instances of the M'B first, and
after the examination of them decide what we will select
for the predicate of our major premise, the inference
will generally be completely fallacious. In short, we
have the rule that the major term P must be decided
upon in advance of the examination of the sample ; and
in like manner in Form IV. (bis) the minor term S must
be decided upon in advance of the drawing.

The same rule follows us into the logic of induction
and hypothesis. If in sampling any class, say the M's,
we first decide what the character P is for which we
propose to sample that class, and also how many instan
ces we propose to draw, our inference is really made
before these latter are drawn, that the proportion of P's
in the whole class is probably about the same as among
the instances that are to be drawn, and the only thing
we have to do is to draw them and observe the ratio.
But suppose we were to draw our inferences without
the predesignation of the character P; then we might in
every case find some recondite character in which those
instances would all agree. That, by the exercise of
sufficient ingenuity, we should be sure to be able to do
this, even if not a single other object of the class M
possessed that character, is a matter of demonstration.
For in geometry a curve may be drawn through any
given series of points, without passing through any one
of another given series of points, and this irrespective of
the number of dimensions. Now, all the qualities of
objects may be conceived to result from variations of a
number of continuous variables ; hence any lot of ob
jects possesses some character in common, not possessed
by any other. It is true that if the universe of quality

A THEORY OF PROBABLE INFERENCE. 163

is limited, this is not altogether true ; but it remains
true that unless we have some special premise from
which to infer the contrary, it always may be possible
to assign some common character of the instances £', S",
S'", etc., drawn at random from among the M' s, which
does not belong to the M'a generally. So that if the
character P were not predesignate, the deduction of
which our induction is the apagogical inversion would
not be valid ; that is to say, we could not reason that if
the M'B did not generally possess the character P, it
would not be likely that the £'s should all possess this
character.

I take from a biographical dictionary the first five
names of poets, with their ages at death. They are,

Aagard, died at 48.

Abeille," " " 76.

Abulola, « " 84.

Abunowas, " " 48.

Accords, " " 45.

These five ages have the following characters in com
mon : —

1. The difference of the two digits composing the
number, divided by three, leaves a remainder of one.

2. The first digit raised to the power indicated by the
second, and then divided by three, leaves a remainder of
one.

3. The sum of the prime factors of each age, including
one as a prime factor, is divisible by three.

Yet there is not the smallest reason to believe that the
next poet's age would possess these characters.

Here we have a conditio sine qud non of valid induc
tion which has been singularly overlooked by those who
have treated of the logic of the subject, and is very fre-

164 A THEORY OF PEOBABLE INFERENCE.

quently violated by those who draw inductions. So ac
complished a reasoner as Dr. Lyon Playfair, for instance,
has written a paper of which the following is an abstract.
He first takes the specific gravities of the three allotropic
forms of carbon, as follows : —

Diamond, 3.48
Graphite, 2.29
Charcoal, 1.88

He now seeks to find a uniformity connecting these three
instances; and he discovers that the atomic weight of
carbon, being 12,

Sp. gr. diamond nearly = 3.46 —

" " graphite " = 2.29 = y!2
" " charcoal « = 1.86 = j/12

This, he thinks, renders it probable that the specific
gravities of the allotropic forms of other elements would,
if we knew them, be found to equal the different roots of
their atomic weight. But so far, the character in which
the instances agree not having been predesignated, the
induction can serve only to suggest a question, and ought
not to create any belief. To test the proposed law, he
selects the instance of silicon, which like carbon exists
in a diamond and in a graphitoidal condition. He finds
for the specific gravities —

Diamond silicon, 2.47
. Graphite silicon, 2.33.*

* The author ought to have noted that this number is open to some
doubt, since the specific gravity of this form of silicon appears to vary
largely. If a different value had suited the theory better, he might have
been able to find reasons for preferring that other value. But I do not
mean to imply that Dr. Playfair has not dealt with perfect fairness with
his facts, except as to the fallacy which I point out.

A THEORY OF PROBABLE INFERENCE. 165

Now, the atomic weight of silicon, that of carbon being
12, can only be taken as 28. But 2.47 does not approx
imate to any root of 28. It is, however, nearly the
cube root of 14, (<\X-i- X 28 = 2.41), while 2.33 is nearly
the fourth root of 28 (v"28 = 2.30). Dr. Playfair claims
that silicon is an instance satisfying his formula. But
in fact this instance requires the formula to be modified ;
and the modification not being predesignate, the instance
cannot count. Boron also exists in a diamond and a
graphitoidal form ; and accordingly Dr. Playfair takes
this as his next example. Its atomic weight is 10.9, and
its specific gravity is 2.68 ; which is the square root of
f X 10.9. There seems to be here a further modification
of the formula not predesignated, and therefore this in
stance can hardly be reckoned as confirmatory. The
next instances which would occur to the mind of any
chemist would be phosphorus and sulphur, which exist
in familiarly known allotropic forms. Dr. Playfair ad
mits that the specific gravities of phosphorus have no
relations to its atomic weight at all analogous to those
of carbon. The different forms of sulphur have nearly
the same specific gravity, being approximately the fifth
root of the atomic weight 32. Selenium also has two
.allotropic forms, whose specific gravities are 4.8 and 4.3 ;
one of these follows the law, while the other does not.
For tellurium the law fails altogether ; but for bromine
and iodine it holds. Thus the number of specific gravi
ties for which the law was predesignate are 8 ; namely,
2 for phosphorus, 1 for sulphur, 2 for selenium, 1 for
tellurium, 1 for bromine, and 1 for iodine. The law
holds for 4 of these, and the proper inference is that
about half the specific gravities of metalloids are roots
of some simple ratio of their atomic weights.

Having thus determined this ratio, we proceed to

166 A THEORY OF PEOBABLE INFERENCE.

inquire whether an agreement half the time with the
formula constitutes any special connection between the
specific gravity and the atomic weight of a metalloid.
As a test of this, let us arrange the elements in the order
of their atomic weights, and compare the specific gravity
of the first with the atomic weight of the last, that of
the second with the atomic weight of the last but one,
and so on. The atomic weights are —

Boron, 10.9 Tellurium, 128.1

Carbon, 12.0 Iodine, 126.9

Silicon, 28.0 Bromine, 80.0

Phosphorus, 31.0 Selenium, 79.1

Sulphur, 32.

There are three specific gravities given for carbon, and
two each for silicon, phosphorus, and selenium. The
question, therefore, is, whether of the fourteen specific
gravities as many as seven are in Playfair's relation
with the atomic weights, not of the same element, but
of the one paired with it. Now, taking the original
formula of Playfair we find

Sp.

gr.

boron

= 2.68

^Te

= 2

.64

Sp.

gr.

carbon

= 1.88

/V/I

= 1

.84

Sp.

gr.

carbon

= 2.29

= 2

.24

1st

Sp.

gr.

phosphorus

= 1.83

^/Se

= 1

.87

Sp.

gr.

phosphorus

= 2.10

'V/Se

= 2

.07

or five such relations without counting that of sulphur
to itself. Next, with the modification introduced by Play-
fair, we have

1st Sp. gr. silicon = 2.47 $% X Br = 2.51

2d Sp. gr. silicon = 2.33 A/2 X Br = 2.33

Sp. gr. iodine = 4.95 ^2x0= 4.90

1st Sp. gr. carbon = 3.48 ^ X I = 3.48

A THEORY OF PROBABLE INFERENCE. 167

It thus appears that there is no more frequent agree
ment with Playfair's proposed law than what is due to
chance.1

Another example of this fallacy was " Bode's law " of
the relative distances of the planets, which was shattered
by the first discovery of a true planet after its enuncia
tion. In fact, this false kind of induction is extremely
common in science and in medicine.2 In the case of
hypothesis, the correct rule has often been laid down ;
namely, that a hypothesis can only be received upon the
ground of its having been verified by successful prediction.
The term predesignation used in this paper appears to be
more exact, inasmuch as it is not at all requisite that the
ratio p should be given in advance of the examination of
the samples. Still, since p is equal to 1 in all ordinary
hypotheses, there can be no doubt that the rule of pre
diction, so far as it goes, coincides with that here laid
down.

We have now to consider an important modification of
the rule. Suppose that, before sampling a class of objects,
we have predesignated not a single character but n char
acters, for which we propose to examine the samples.
This is equivalent to making n different inductions from
the same instances. The probable error in this case is
that error whose probability for a simple induction is only
(|)n, and the theory of probabilities shows that it in-

1 As the relations of the different powers of the specific gravity would
be entirely different if any other substance than water were assumed as
the standard, the law is antecedently in the highest degree improbable.
This makes it likely that some fallacy was committed, but does not show
what it was.

2 The physicians seem to use the maxim that you cannot reason from
post hoc to propter hoc to mean (rather obscurely) that cases must not be
used to prove a proposition that has only been suggested by these cases
themselves.

168 A THEOEY OF PEOBABLE INFEEENCE.

creases but slowly with n ; in fact, for n — 1000 it is only
about five times as great as for n = 1, so that with only
25 times as many instances the inference would be as
secure for the former value of n as with the latter ; with
100 times as many instances an induction in which n —
10,000,000,000 would be equally secure. Now the whole
universe of characters will never contain such a number
as the last ; and the same may be said of the universe of
objects in the case of hypothesis. So that, without any
voluntary predesignation, the limitation of our imagina
tion and experience amounts to a predesignation far
within those limits ; and we thus see that if the number
of instances be very great indeed, the failure to predes-
ignate is not an important fault. Of characters at all
striking, or of objects at all familiar, the number will
seldom reach 1,000 ; and of very striking characters or
very familiar objects the number is still less. So that if
a large number of samples of a class are found to have
some very striking character in common, or if a large
number of characters of one object are found to be pos
sessed by a very familiar object, we need not hesitate to
infer, in the first case, that the same characters belong
to the whole class, or, in the second case, that the two
objects are practically identical ; remembering only that
the inference is less to be relied upon than it would be
had a deliberate predesignation been made. This is no
doubt the precise significance of the rule sometimes laid
down, that a hypothesis ought to be simple, — simple
here being taken in the sense of familiar.

This modification of the rule shows that, even in the
absence of voluntary predesignation, some slight weight
is to be attached to an induction or hypothesis. And
perhaps when the number of instances is not very small,
it is enough to make it worth while to subject the in-

A THEORY OF PROBABLE INFERENCE. 169

ference to a regular test. But our natural tendency will
be to attach too much importance to sucli suggestions,
and we shall avoid waste of time in passing them by
without notice until some stronger plausibility presents
itself.

IX.

In almost every case in which we make an induction
or a hypothesis, we have some knowledge which renders
our conclusion antecedently likely or unlikely. The ef
fect of such knowledge is very obvious, and needs no
remark. But what also very often happens is that we
have some knowledge, which, though not of itself bearing
upon the conclusion of the scientific argument, yet serves
to render our inference more or less probable, or even
to alter the terms of it. Suppose, for example, that we
antecedently know that all the M's strongly resemble
one another in regard to characters of a certain order.
Then, if we find that a moderate number of M9s taken
at random have a certain character, P, of that order, we
shall attach a greater weight to the induction than we
should do if we had not that antecedent knowledge.
Thus, if we find that a certain sample of gold has a
certain chemical character, — since we have very strong
reason for thinking that all gold is alike in its chemical
characters, — we shall have no hesitation in extending
the proposition from the one sample to gold in general.
Or if we know that among a certain people, — say the
Icelanders, — an extreme uniformity prevails in regard
to all their ideas, then, if we find that two or three in
dividuals taken at random from among them have all
any particular superstition, we shall be the more ready
to infer that it belongs to the whole people from what
we know of their uniformity. The influence of this sort

170 A THEORY OF PKOBABLE INFERENCE.

of uniformity upon inductive conclusions was strongly in
sisted upon by Philodemus, and some very exact concep
tions in regard to it may be gathered from the writings
of Mr. Galton. Again, suppose we know of a certain
character, P, that in whatever classes of a certain des
cription it is found at all, to those it usually belongs as
a universal character ; then any induction which goes
toward showing that all the M's are P will be greatly
strengthened. Thus it is enough to find that two or
three individuals taken at random from a genus of ani
mals have three toes on each foot, to prove that the same
is true of the whole genus ; for we know that this is a
generic character. On the other hand, we shall be slow
to infer that all the animals of a genus have the same
color, because color varies in almost every genus. This
kind of uniformity seemed to J. S. Mill to have so con
trolling an influence upon inductions, that he has taken
it as the centre of his whole theory of the subject.

Analogous considerations modify our hypothetic infer
ences. The sight of two or three words will be sufficient
to convince me that a certain manuscript was written by
myself, because I know a certain look is peculiar to it.
So an analytical chemist, who wishes to know whether a
solution contains gold, will be completely satisfied if it
gives a precipitate of the purple of cassius with chloride
of tin ; because this proves that either gold or some hith
erto unknown substance is present. These are examples
of characteristic tests. Again, we may know of a certain
person, that whatever opinions he holds he carries out
with uncompromising rigor to their utmost logical con
sequences ; then, -if we find his views bear some of the
marks of any ultra school of thought, we shall readily
conclude that he fully adheres to that school.

There are thus four different kinds of uniformity and

A THEORY OF PROBABLE INFERENCE. 171

non-uniformity which may influence our ampliative in
ferences : —

1. The members of a class may present a greater or
less general resemblance as regards a certain line of char
acters.

2. A character may have a greater or less tendency
to be present or absent throughout the whole of whatever
classes of certain kinds.

3. A certain set of characters may be more or less
intimately connected, so as to be probably either present
or absent together in certain kinds of objects.

4. An object may have more or less tendency to
possess the whole of certain sets of characters when it
possesses any of them.

A consideration of this sort may be so strong as to
amount to demonstration of the conclusion. In this case,
the inference is mere deduction, — that is, the application
of a general rule already established. In other cases, the
consideration of uniformities will not wholly destroy the
inductive or hypothetic character of the inference, but
will only strengthen or weaken it by the addition of a
new argument of a deductive kind.

We have thus seen how, in a general way, the processes
of inductive and hypothetic inference are able to afford
answers to our questions, though these may relate to
matters beyond our immediate ken. In short, a theory
of the logic of verification has been sketched out. This
theory will have to meet the objections of two opposing
schools of logic.

The first of these explains induction by what is called
the doctrine of Inverse Probabilities, of which the follow-

172 A THEORY OF PROBABLE INFERENCE.

ing is an example : Suppose an ancient denizen of the
Mediterranean coast, who had never heard of the tides,
had wandered to the shore of the Atlantic Ocean, and
there, on a certain number m of successive days had
witnessed the rise of the sea. Then, says Quetelet, he
would have been entitled to conclude that there was a
probability equal to ^ t_ that the sea would rise on the
next following day.1 Putting m = 0, it is seen that
this view assumes that the probability of a totally un
known event is £; or that of all theories proposed for
examination one half are 4;rue. In point of fact, we
know that although theories are not proposed unless
they present some decided plausibility, nothing like one
half turn out to be true. But to apply correctly the
doctrine of inverse probabilities, it is necessary to know
the antecedent probability of the event whose proba
bility is in question. Now, in pure hypothesis or induc
tion, we know nothing of the conclusion antecedently
to the inference in hand. Mere ignorance, however,
cannot advance us toward any knowledge ; therefore it
is impossible that the theory of inverse probabilities
should rightly give a value for the probability of a pure
inductive or hypothetic conclusion. For it cannot do
this without assigning an antecedent probability to this
conclusion ; so that if this antecedent probability rep
resents mere ignorance (which never aids us), it cannot
do it at all.

The principle which is usually assumed by those who
seek to reduce inductive reasoning to a problem in in
verse probabilities is, that if nothing whatever is known
about the frequency of occurrence of an event, then any
one frequency is as probable as any other. But Boole

1 See Laplace, "Theorie Analitique des Probabilites," livre ii. chap. vi.

A THEORY OF PROBABLE INFERENCE. 173

has shown that there is no reason whatever to prefer this
assumption, to saying that any one " constitution of the
universe" is as probable as any other. Suppose, for
instance, there were four possible occasions upon which
an event might occur. Then there would be 16 " con
stitutions of the universe," or possible distributions of
occurrences and non-occurrences. They are shown in
the following table, where Y stands for an occurrence
and N for a non-occurrence.

4 occurrences.

3 occurrences.

2 occurrences.

1 occurrence.

0 occurrence.

YYYY

YYYN

YYNN

YNNN

NNNN

YYNY

YNYN

NYNN

YNYY

YNNY

NNYN

NYYY

NYYN

NNNT

NYNY

NNYY

It will be seen that different frequencies result some
from more and some from fewer different " constitutions
of the universe," so that it is a very different thing to
assume that all frequencies are equally probable from
what it is to assume that all constitutions of the universe
are equally probable.

Boole says that one assumption is as good as the other.
But I will go further, and say that the assumption that
all constitutions of the universe are equally probable is
far better than the assumption that all frequencies are
equally probable. For the latter proposition, though it
may be applied to any one unknown event, cannot be
applied to all unknown events without inconsistency.
Thus, suppose all frequencies of the event whose occur
rence is represented by I^in the above table are equally
probable. Then consider the event which consists in a
Y following a Y or an N following an N. The possible

174 A THEORY OF PROBABLE INFERENCE.

ways in which this event may occur or not are shown in
the following table : —

3 occurrences.

YYYY

NNNN

2 occurrences,

YYYN

NNNY

1 occurrence.

YYNY

NNYN

0 occurrence.

YNYN
NYNY

YYNN
NNYY

YNNY
NYYN

N YYY

YNNN

YNYY

NYNN

It will be found that assuming the different frequencies
of the first event to be equally probable, those of this new
event are not so, — the probability of three occurrences
being half as large again as that of two, or one. On the
other hand, if all constitutions of the universe are equally
probable in the one case, they are so in the other ; and
this latter assumption, in regard to perfectly unknown
events, never gives rise to any inconsistency.

Suppose, then, that we adopt the assumption that any
one constitution of the universe is as probable as any
other ; how will the inductive inference then appear, con
sidered as a problem in probabilities ? The answer is
extremely easy ; 1 namely, the occurrences or non-occur
rences of an event in the past in no way affect the proba
bility of its occurrence in the future.

Boole frequently finds a problem in probabilities to be
indeterminate. There are those to whom the idea of an
unknown probability seems an absurdity. Probability,
they say, measures the state of our knowledge, and ig
norance is denoted by the probability |. But I appre
hend that the expression " the probability of an event "
is an incomplete one. A probability is a fraction whose

1 See Boole, "Laws of Thought."

A THEORY OF PROBABLE INFERENCE. 175

numerator is the frequency of a specific kind of event,
while its denominator is the frequency of a genus embrac
ing that species. Now the expression in question names
the numerator of the fraction, but omits to name the de
nominator. There is a sense in which it is true that the
probability of a perfectly unknown event is one half ;
namely, the assertion of its occurrence is the answer to
a possible question answerable by " yes " or " no," and
of all such questions just half the possible answers are
true. But if attention be paid to the denominators of
the fractions, it will be found that this value of J is one
of which no possible use can be made in the calculation
of probabilities.

The theory here proposed does not assign any proba
bility to the inductive or hypothetic conclusion, in the
sense of undertaking to say how frequently that conclu
sion would be found true. It does not propose to look
through all the possible universes, and say in what pro
portion of them a certain uniformity occurs ; such a
proceeding, were it possible, would be quite idle. The
theory here presented only says how frequently, in this
universe, the special form of induction or hypothesis
would lead us right. The probability given by this theory
is in every way different — in meaning, numerical value,
and form — from that of those who would apply to am-
pliative inference the doctrine of inverse chances.

Other logicians hold that if inductive and hypothetic
premises lead to true oftener than to false conclusions,
it is only because the universe happens to have a certain
constitution. Mill and his followers maintain that there
is a general tendency toward uniformity in the universe,
as well as special uniformities such as those which we
have considered. The Abbe* Gratry believes that the
tendency toward the truth in induction is due to a mirac-

176 A THEORY OF PROBABLE INFERENCE.

ulous intervention of Almighty God, whereby we are led
to make such inductions as happen to be true, and are
prevented from making those which are false. Others
have supposed that there is a special adaptation of the
mind to the universe, so% that we are more apt to make
true theories than we otherwise should be. Now, to say
that a theory such as these is necessary to explaining the
validity of induction and hypothesis is to say that these
modes of inference are not in themselves valid, but that
their conclusions are rendered probable by being probable
deductive inferences from a suppressed (and originally
unknown) premise. But I maintain that it has been
shown that the modes of inference in question are neces
sarily valid, whatever the constitution of the universe, so
long as it admits of the premises being true. Yet I am
willing to concede, in order to concede as much as possi
ble, that when a man draws instances at random, all that
he knows is that he tries to follow a certain precept ; so
that the sampling process might be rendered generally
fallacious by the existence of a mysterious and malign
connection between the mind and the universe, such that
the possession by an object of an unperceived character
might influence the will toward choosing it or rejecting
it. Such a circumstance would, however, be as fatal to
deductive as to ampliative inference. Suppose, for exam
ple, that I were to enter a great hall where people were
playing rouge et noir at many tables ; and suppose that
I knew that the red and black were turned up with equal
frequency. Then, if I were to make a large number of
mental bets with myself, at this table and at that. I. might,
by statistical deduction, expect to win about half of them,
— precisely as I might expect, from the results of these
samples, to infer by induction the probable ratio of fre
quency of the turnings of red and black in the long run,

A THEORY OF PROBABLE INFERENCE. 177

if I did not know it. But could some devil look at eacli
card before it was turned, and then influence me mentally
to bet upon it or to refrain therefrom, the observed ratio
in the cases upon which I had bet might be quite different
from the observed ratio in those cases upon which I had
not bet. I grant, then, that even upon my theory some
fact has to be supposed to make induction and hypothe
sis valid processes ; namely, it is supposed that the su
pernal powers withhold their hands and let me alone,
and that no mysterious uniformity or adaptation inter
feres with the action of chance. But then this negative
fact supposed by my theory plays a totally different part
from the facts supposed to be requisite by the logicians
of whom I have been speaking. So far as facts like those
they suppose can have any bearing, they serve as major
premises from which the fact inferred by induction or
hypothesis might be deduced ; while the negative fact
supposed by me is merely the denial of any major premise
from which the falsity of the inductive or hypothetic con
clusion could in general be deduced. Nor is it necessary
to deny altogether the existence of mysterious influences
adverse to the validity of the inductive and hypothetic
processes. So long as their influence were not too over
whelming, the wonderful self-correcting nature of the
ampliative inference would enable us, even if they did
exist, to detect and make allowance for them.

Although the universe need have no peculiar consti
tution to render ampliative inference valid, yet it is worth
while to inquire whether or not it has such a constitu
tion ; for if it has, that circumstance must have its effect
upon all our inferences. It cannot any longer be denied
that the human intellect is peculiarly adapted to the
comprehension of the laws and facts of nature, or at
least of some of them ; and the effect of this adaptation

178 A THEORY OF PROBABLE INFERENCE.

upon our reasoning will be briefly considered in the next
section. Of any miraculous interference by the higher
powers, we know absolutely nothing ; and it seems in
the present state of science altogether improbable. The
effect of a knowledge of special uniformities upon ampli-
ative inferences has already been touched upon. That
there is a general tendency toward uniformity in nature
is not merely an unfounded, it is an absolutely absurd,
idea in any other sense than that man is adapted to his
surroundings. For the universe of marks is only limited
by the limitation of human interests and powers of ob
servation. Except for that limitation, every lot of objects
in the universe would have (as I have elsewhere shown)
some character in common and peculiar to it. Conse
quently, there is but one possible arrangement of charac
ters among objects as they exist, and there is no room
for a greater or less degree of uniformity in nature. If
nature seems highly uniform to us, it is only because our
powers are adapted to our desires.

XI.

The questions discussed in this essay relate to but a
small part of the Logic of Scientific Investigation. Let
us just glance at a few of the others.

Suppose a being, from some remote part of the uni
verse, where the conditions of existence are inconceivably
different from ours, to be presented with a United States
Census Report, — which is for us a mine of valuable in
ductions, so vast as almost to give that epithet a new signi
fication. He begins, perhaps, by comparing the ratio of
indebtedness to deaths by consumption in counties whose
names begin with the different letters of the alphabet.
It is safe to say that he would find the ratio everywhere

B a i T i s

WINTER KA1N FATX

CASchott

A THEOKY OF PROBABLE INFERENCE. 179

the same, and thus his inquiry would lead to nothing.
For an induction is wholly unimportant unless the pro
portions of P's among the M's and among the non-M's
differ ; and a hypothetic inference is unimportant unless
it be found that S has either a greater or a less propor
tion of the characters of M than it has of other charac
ters. The stranger to this planet might go on for some
time asking inductive questions that the Census would
faithfully answer, without learning anything except that
certain conditions were independent of others. At length,
it might occur to him to compare the January rain-fall
with the illiteracy. What he would find is given in the
folio win": table1 : —

REGION.

January Rain-fall.

Illiteracy.

Atlantic Sea-coast, Port-)
land to Washington )

Inches.

0.92

Per cent.
11

Vermont, Northern and)
Western New York )

0.78

Upper Mississippi E/iver .

0.52

Ohio River Valley . . .

0.74

Lower Mississippi, Red)
River, and Kentucky )

1.08

Mississippi Delta and)
Northern Gulf Coast )

1.09

Southeastern Coast . . .

0.68

1 The different regions with the January rain-fall are taken from Mr.
Schott's work. The percentage of illiteracy is roughly estimated from the
numbers given in the Keport of the 1870 Census.

180 A THEORY OF PROBABLE INFERENCE.

He would infer that in places that are drier in January
there is, not always but generally, less illiteracy than
in wetter places. A detailed comparison between Mr.
Schott's map of the winter rain-fall with the map of
illiteracy in the general census, would confirm the result
that these two conditions have a partial connection.
This is a very good example of an induction in which
the proportion of P's among the M9s is different, but
not very different, from the proportion among the non-
Jf's. It is unsatisfactory ; it provokes further inquiry ;
we desire to replace the M by some different class, so
that the two proportions may be more widely separated.
Now we, knowing as much as we do of the effects of
winter rain-fall upon agriculture, upon wealth, etc., and
of the causes of illiteracy, should come to such an inquiry
furnished with a large number of appropriate conceptions ;
so that we should be able to ask intelligent questions not
unlikely to furnish the desired key to the problem. But
the strange being we have imagined could only make his
inquiries hap-hazard, and could hardly hope ever to find
the induction of which he was in search.

Nature is a far vaster and less clearly arranged reper
tory of facts than a census report ; and if men had not
come to it with special aptitudes for guessing right, it
may well be doubted whether in the ten or twenty thou
sand years that they may have existed their greatest
mind would have attained the amount of knowledge
which is actually possessed by the lowest idiot. But,
in point of fact, not man merely, but all animals derive
by inheritance (presumably by natural selection) two
classes of ideas which adapt them to their environment.
In the first place, they all have from. birth some notions,
however crude and concrete, of force, matter, space, and
time ; and, in the next place, they have some notion of

A THEORY OF PROBABLE INFERENCE. 181

what sort of objects their fellow-beings are, and of how
they will act on given occasions. Our innate mechanical
ideas were so nearly correct that they needed but slight
correction. The fundamental principles of statics were
made out by Archimedes. Centuries later Galileo began
to understand the laws of dynamics, which in our times
have been at length, perhaps, completely mastered. The
other physical sciences are the results of inquiry based
on guesses suggested by the ideas of mechanics. The
moral sciences, so far as they can be called sciences,
are equally developed out of our instinctive ideas about
human nature. Man has thus far not attained to any
knowledge that is not in a wide sense either mechanical
or anthropological in its nature, and it may be reasonably
presumed that he never will.

Side by side, then, with the well established propo
sition that all knowledge is based on experience, and
that science is only advanced by the experimental verifi
cations of theories, we have to place this other equally
important truth, that all human knowledge, up to the
highest flights of science, is but the development of our
inborn animal instincts.

NOTE A.

BOOLE, De Morgan, and their followers, frequently
speak of a " limited universe of discourse " in logic. An
unlimited universe would comprise the whole realm of the
logically possible. In such a universe, every universal
proposition, not tautologous, is false ; every particular
proposition, not absurd, is true. Our discourse seldom
relates to this universe : we are either thinking of the
physically possible, or of the historically existent, or of
the world of some romance, or of some other limited
universe.

But besides its universe of objects, our discourse also
refers to a universe of characters. Thus, we might
naturally say that virtue and an orange have nothing
in common. It is true that the English word for each
is spelt with six letters, but this is not one of the marks
of the universe of our discourse.

A universe of things is unlimited in which every com
bination of characters, short of the whole universe of
characters, occurs in some object. In like manner, the
universe of characters is unlimited in case every aggre
gate of things short of the whole universe of things
possesses in common one of the characters of the uni
verse of characters. The conception of ordinar}^ syllo
gistic is so unclear that it would hardly be accurate to
say that it supposes an unlimited universe of characters ;

ON A LIMITED UNIVERSE OF MARKS. 183

but it comes nearer to that than to any other consistent
view. The non-possession of any character is regarded
as implying the possession of another character the nega
tive of the first.

In our ordinary discourse, on the other hand, not only
are both universes limited, but, further than that, we
have nothing to do with individual objects nor simple
marks ; so that we have simply the two distinct universes
of things and marks related to one another, in general, in
a perfectly indeterminate manner. The consequence is,4
that a proposition concerning the relations of two groups
of marks is not necessarily equivalent to any proposition
concerning classes of things ; so that the distinction
between propositions in extension and propositions in
comprehension is a real one, separating two kinds of
facts, whereas in the view of ordinary syllogistic the
distinction only relates to two modes of considering any
fact. To say that every object of the class S is included
among the class of P's, of course must imply that every
common character of the P's is a common character of
the $'s. But the converse implication is by no means
necessary, except with an unlimited universe of marks.
The reasonings in depth of which I have spoken, suppose,
of course, the absence of any general regularity about the
relations of marks and things.

I may mention here another respect in which this view
differs from that of ordinary logic, although it is a point
which has, so far as I am aware, no bearing upon the
theory of probable inference. It is that under this view
there are propositions of which the subject is a class of
things, while the predicate is a group of marks. Of such
propositions there are twelve species, distinct from one
another in the sense that any fact capable of being ex
pressed by a proposition of one of these species cannot

184 ON A LIMITED UNIVERSE OF MARKS.

be expressed by any proposition of another species. The
following are examples of six of the twelve species : —

1. Every object of the class S possesses every character of
the group TT.

2. Some object of the class S possesses all characters of
the group TT.

3. Every character of the group TT is possessed by some
object of the class S.

4. Some character of the group TT is possessed by all the
objects of the class S.

5. Every object of the class S possesses some character of
the group TT.

6. Some object of the class S possesses some character of
the group TT.

The remaining six species of propositions are like the
above, except that they speak of objects wanting charac
ters instead of possessing characters.

But the varieties of proposition do not end here ; for
we may have, for example, such a form as this : " Some
object of the class S possesses every character not want
ing to any object of the class P." In short, the relative
term " possessing as a character," or its negative, may
enter into the proposition any number of times. We
may term this number the order of the proposition.

An important characteristic of this kind of logic is the
part that immediate inference plays in it. Thus, the
proposition numbered 3, above, follows from No. 2, and
No. 5 from No. 4. It will be observed that in both cases
a universal proposition (or one' that states the non-
existence of something) follows from a particular propo
sition (or one that states the existence of something).
All the immediate inferences are essentially of that
nature. A particular proposition is never immediately
inferable from a universal one. (It is true that from

ON A LIMITED UNIVERSE OF MARKS. 185

" no A exists " we can infer that " something not A
exists ; " but this is not properly an immediate infer
ence, — it really supposes the additional premise that
u something exists.") There are also immediate in
ferences raising and reducing 'the order of propositions.
Thus, the proposition of the second order given in the
last paragraph follows from " some S is a P." On the
other hand, the inference holds, —

Some common character of the S's is wanting to every
thing except _P's ;
.'. Every S is a P.

The necessary and sufficient condition of the existence
of a syllogistic conclusion from two premises is simple
enough. There is a conclusion if, and only if, there is
a middle term distributed in one premise and undistribu
ted in the other. But the conclusion is of the kind called
spurious l by De Morgan if, and only if, the middle term
is affe'cted by a " some " in both premises. For exam
ple, let the two premises be, —

Every object of the class /S wants some character of the
group p.;

Every object of the class P possesses some character not of
the group //,.

The middle term /JL is distributed in the second premise,
but not in the first ; so that a conclusion can be drawn.
But, though both propositions are universal, ^ is under
a " some " in both ; hence only a spurious conclusion
can be drawn, and in point of fact we can infer both of
the following : —

1 On spurious propositions, see Mr. B. I. Oilman's paper in the Johns
Hopkins University Circular for August, 1882. The number of such
forms in any order is probably finite.

186 ON A LIMITED UNIVERSE OF MARKS.

Every object of the class S wants a character other than
some character common to the class P ;

Every object of the class P possesses a character other
than some character wanting to every object of the class S.

The order of the conclusion is always the sum of the
orders of the premises ; but to draw up a rule to deter
mine precisely what the conclusion is, would be difficult.
It would at the same time be useless, because the prob
lem is extremely simple when considered in the light of
the logic of relatives.

NOTE B.

A DUAL relative term, such as " lover," " benefactor,"
" servant," is a common name signifying a pair of ob
jects. Of the two members of the pair, a determinate
one is generally the first, and the other the second ; so
that if the order is reversed, the pair is not considered as
remaining the same.

Let A, B, C, D, etc., be all the individual objects in
the universe ; then all the individual pairs may be arrayed
in a block, thus : —

A:A A:B A:C A : D etc.
B:A B:B B:C B : D etc.
C:A C:B C:C C:D etc.
D:A D : B D : C D : D etc.
etc. etc. etc. etc. etc.

A general relative may be conceived as a logical aggre
gate of a number of such individual relatives. Let I de
note " lover ; " then we may write

where (Z)# is a numerical coefficient, whose value is 1 in
case I is a lover of J, and 0 in the opposite case, and
where the sums are to be taken for all individuals in the

unverse.

188 THE LOGIC OF EEL ATI YES.

Every relative term has a negative (like any other
term) which may be represented by drawing a straight
line over the sign for the relative itself. The negative
of a relative includes every pair that the latter excludes,
and vice versa. Every relative has also a converse, pro
duced by reversing the order of the members of the pair.
Thus, the converse of u lover" is "loved." The con
verse may be represented by drawing a curved line over
the sign for the relative, thus : I. It is defined by the
equation

The following formulae are obvious, but important : —

(i -< b) = (l -< i) (i -< b) = (l-< b).

Relative terms can be aggregated and compounded like
others. Using -f for the sign of logical aggregation, and
the comma for the sign of logical composition (Boole's
multiplication, here to be called non-relative or internal
multiplication), we have the definitions

The first of these equations, however, is to be understood
in a peculiar way : namely, the + in the second member
is not strictly addition, but an operation by which

Instead of (l)$ + (&)y- , we might with more accuracy
write

THE LOGIC OF EELATIVES. 189

The main formulas of aggregation and composition are

( If I -< s and b -< s, then I + b — < s. |_
(If s — < I and s -< b, then s-< £,&. I

( If Z + £ -< 5, then Z -< s and £ -< s. \
(If 5-< Z,#, then s — < Z and s -< &. )

( (I + &) ?s -< l,s + &,$• )
1 (I + s),(b + s) -< lt b + s. )

The subsidiary formulas need not be given, being the
same as in non-relative logic.

We now come to the combination of relatives. Of
these, we denote two by special symbols ; namely, we

write

lb for lover of a benefactor,
and

I f b for lover of everything hut benefactors.

The former is called a particular combination, because
it implies the existence of something loved by its relate
and a benefactor of its correlate. The second combina
tion is said to be universal, because it implies the non-
existence of anything except what is either loved by its
relate or a benefactor of its correlate. The combination
lb is called a relative product, / f b a relative sum. The
I and b arc said to be undistributed in both, because if
I - C s, then lb - C sb and I f b ^< s f b ; and if b -< 5,
then lb - C Is and I f b -< I f .9.

The two combinations are defined by the equations

The sign of addition in the last formula has the same
signification as in the equation defining non-relative
multiplication.

190 THE LOGIC OF KELATIYES.

Relative addition and multiplication are subject to the
associative law. That is,

l(bs) =

Two formulae so constantly used that hardly anything
can be done without them are

The former asserts that whatever is lover of an object
that is benefactor of everything but a servant, stands to
everything but servants in the relation of lover of a
benefactor. The latter asserts that whatever stands to
any servant in the relation of lover of everything but its
benefactors, is a lover of everything but benefactors of
servants. The following formulas are obvious and triv
ial:—

Is + Is -< (l+b)s

z,&t«-<(*t«)»(at«)-

Unobvious and important, however, are these : —
(I + b) s -< Is + bs

(Jt*),(&t*)-<MU

There are a number of curious development formulae.
Such are

(I + b) t s = 2P {\l f (s

n (b + s) = 2P{\_(1 +P) t b-],[_(l +p) f s-]}.

The summations and multiplications denoted by ^ and IT
are to be taken non-relatively, and all relative terms are
to be successively substituted for p.

THE LOGIC OF RELATIVES. 191

The negatives of the combinations follow these rules :

I I b = Ib I b = I ~f b

The converses of combinations are as follows : —

Individual dual relatives are of two types, —
A : A and A : B.

Relatives containing no pair of an object with itself are
called alio-relatives as opposed to self -relatives. The
negatives of alio-relatives pair every object with itself.
Relatives containing no pair of an object with anything
but itself are called concurrents as opposed to opponents.
The negatives of concurrents pair every object with every
other.

There is but one relative which pairs every object with
itself and with every other. It is the aggregate of all
pairs, and is denoted by GO. It is translated into ordi
nary language by " coexistent with." Its negative is 0.
There is but one relative which pairs every object with
itself and none with any other. It is

(A : A) + (B : B) + (C : C) + etc. ;

is denoted by 1, and in ordinary language is "identical
with — ." Its negative, denoted by n, is " other than—,"
or " not."

No matter what relative term x may be, we have

0 -< x x -< oo.

192 THE LOGIC OF RELATIVES.

Hence, obviously

x + 0 = x x, GO = x

a?-foo— OQ cc , 0 = 0.

The last formula hold for the relative operations ; thus,

# f GO := 00 XO = 0.

GO f sc = oo 0 a? — 0.

The formulas

X + 0 = X X, GO = X

also hold if we substitute the relative operations, and
also 1 for oo, and n for 0 ; thus,

x f n = x x~L = x.

n -f x = x \x = x.
We have also

l + l=o, 1,1 = 0.

To these partially correspond the following pair of highly
important formulas : —

1 -< 1 1 1 l~l -< n.

The logic of relatives is highly multiform ; it is char
acterized by innumerable immediate inferences, and by
various distinct conclusions from the same sets of premi
ses. An example of the first character is afforded by
Mr. Mitchell's Flv following from Flv,. As an instance
of the second, take the premises,

Every man is a lover of an animal ;
and

Every woman is a lover of a non-animal.

From these we can equally infer that

Every man is a lover of something which stands to each
woman in the relation of not being the only thing loved
by her,

THE LOGIC OF RELATIVES. 193

and that

Every woman is a lover of something which stands to
each man in the relation of not being the only thing loved
by him.

The effect of these peculiarities is that this algebra can
not be subjected to hard and fast rules like those of
the Boolian calculus ; and all that can be done in this
place is to give a general idea of the way of working with
it. The student must at the outset disabuse himself of
the notion that the chief instruments of algebra are the
inverse operations. General algebra hardly knows any
inverse operations. When an inverse operation is iden
tical with a direct operation with an inverse quantity
(as subtraction is the addition of the negative, and as
division is multiplication by the reciprocal), it is useful ;
otherwise it is almost always useless. In ordinary alge
bra, we speak of the " principal value " of the logarithm,
etc., which is a direct operation substituted for an in
definitely ambiguous inverse operation. The elimination
and transposition in this algebra really does depend,
however, upon formulae quite analogous to the

x + (- x) = 0 x X \ = 1,

of arithmetical algebra. These formulas are

I + 1 = oo 1 -< 1 1 .

For example, to eliminate * from the two propositions
1-C Is l-< sb,

we relatively multiply them in such an order as to bring
the two s's together, and then apply the second of the
above formulas, thus : —

1 -< IsSb -< l\\b.

194 THE LOGIC OF EELATIVES.

This example shows the use of the association formulae
in bringing letters together. Other formulas of great
importance for this purpose are

The distribution formula are also useful for this pur
pose.

When the letter to be eliminated has thus been re
placed by one of the four relatives, — 0, GO, 1, n, — the
replacing relative can often be got rid of by means of
one of the formulae

When we have only to deal with universal propositions,

.it will be found convenient so to transpose everything

from subject to predicate as to make the subject L Thus,

if we have given I -< 6, we may relatively add I to both

sides ; whereupon we have

Every proposition will then be in one of the forms
1 -< b 1 1 l-<bl.

With a proposition of the form 1 — <^ b f ?, we have the
right (1) to transpose the terms, and (2) to convert the
terms. Thus, the following are equivalent : —

1 -< b 1 1

1 -< ?t ft-

With a proposition of the form 1 -< b I, we have only
the right to convert the predicate giving 1 -< I b.

THE LOGIC OF RELATIVES. 195

With three terms, there are four forms of universal
propositions, namely : —

Of these, the third is an immediate inference from the
second.

By way of illustration, we may work out the syllo
gisms whose premises are the propositions of the first
order referred to in Note A. Let a and c be class terms,
and let {3 be a group of characters. Let p he the relative
" possessing as a character." The non-relative terms
are to be treated as relatives, — a, for instance, being
considered as " a coexistent with " and a as " coexistent
with a that is." Then, the six forms of affirmative
propositions of the first order are

The various kinds of syllogism are as follows : —
1. Premises : 1 -< a f p f /3 1 -< c ^ p t /?•
Convert one of the premises and multiply,

The treatment would be the same if one or both of
the premises were negative ; that is, contained p in place
of p.

196 THE LOGIC OF RELATIVES.

2. Premises : 1 -< a ^p t /? 1 -< c (p f fi).
We have

The same with negatives.

3. Premises : 1 -< a ( p f j3) 1 -< % (p t /•?).

1 -< « (JP t P) (£t£) c -<
The same with negatives.

4. Premises : 1 -< « f.p f /? 1 -< c

If one of the premises, say the first, were negative, we
should obtain a similar conclusion, —

but from this again jt? could be eliminated, giving
1 -< a f c, or a -< c.

5. Premises : 1 -< « (p t /3) 1 -< (c t^?) y^.

1 -< o (p t/3)^ (^ t 0 -< «-P (^ t c).
If either premise were negative, ^> could be eliminated,
giving 1 ^^ « 0, or some a is c.

6. Premises : 1 -< (a t^) /? 1 -< (c ^ p) ft.

7. Premises : 1 -< a f.P t

l-< («f^t/5)(

8. Premises : 1 -< a (p -\ j3) l-<cp1[

9. Premises : 1 -< (a f ^) y8 1 -< gp t A
1 -< («tjP))8 (jSt^c) -<

THE LOGIC OF RELATIVES. 197

If one premise is negative, we have the further conclu
sion 1 —<[ dc.

10. Premises : 1 -< ap f ft 1 -< cp

1 -< (ap t /?) (j8 t $ c) -< «.p t j><?.

11. Premises : I -< a^p-fft 1 -<

We might also conclude

but this conclusion is an immediate inference from the
other ; for

If one premise is negative, we have the further conclu
sion 1 -< a f c.

12. Premises: 1 -< a (j? | /*) l
1 -< «(^t)8) 08j> t<0 -<
If one premise is negative, we have the further inference

13. Premises : l-<(a^p)(3 1 -< £ f ^ A

1 -< (« t^) /5 (^ t c) -< ("v t^) («^ t c).
14 Premises: 1 -< ap-\ ft 1 -< c-fp/3.

If one premise is negative, we have the further spurious
inference 1 -<^ a n f <?.

15. Premises: l-<

1 -< (« t^« ( t 0 -< « tl> (u/> t
We can'al&o infer 1 -< (a f^)^ t c-

198 THE LOGIC OF RELATIVES.

16. Premises : 1 -< a ^p f (3 1 -< cpp.

If one premise is negative, we can further infer

17. Premises : 1 -< a (p f /?) l-<cpp.

1 -< a (p t P) fipc -< appc.
If one premise is negative, we have the further spurious

conclusion 1 — <[ a lie.

18. Premises: 1 -< (a^p~)p 1 -<

19. Premises: 1 -< ap} (3 l-<cp(3.

l-< (ap-tP)jtpc-< appc.
If one premise is negative, we further conclude 1

20. Premises : 1 -< a -\p p l-<cpp.

21. Premises: 1 -<

When we have to do with particular propositions, we
have the proposition oo -<[ 0, or "something exists;"
for every particular proposition implies this. Then every-
proposition can be put into one or other of the four
forms

oo -< 0-j^tO

00 -< (0 t 0 00

oo -< (0 1 1 oo

OO -< 00 I 00.

Each of these propositions immediately follows from the
one above it. The enveloped expressions which form the

THE LOGIC OF EELATIYES. 199

predicates have the remarkable property that each is
either 0 or oo. This fact gives extraordinary freedom
in the use of the formulas. In particular, since if any
thing not zero is included under such an expression, the
whole universe is included, it will be quite unnecessary
to write the GO -<^ which begins every proposition.

Suppose that / and g are general relatives signifying
relations of things to times. Then, Dr. Mitchell's six
forms of two dimensional propositions appear thus : —

^ttv = oo/oo.
It is obvious that I f 0 -< Z, for

Z|0-< (7-j-O) oo -< ZfO oo-< Jttt-< l

If then we have Of/fO as one premise, and the other
contains g, we may substitute for g the product (/, g).

g -< 0r, oo -< g, (0 f/t 0) -< g,f.
From the two premises

oo (/t 0) and 0 f g oo,
by the application of the formulas

we have

These formulae give the first column of Dr. Mitchell's
rule on page 90.

200 THE LOGIC OF KELATIVES.

The following formulae may also be applied

2. (Ot/)oo

3. (Of/)oc

4. (Of/)Gc (0 | S) oc -< (0 t/)yo,

5. (Ot/tO)(Ot0oo) = Of (#/,/) t

6. (Ot/)oo (Of #00) = (0t <//,/) oo.

9- (Ot/«>),(0tflroo) =0f/oo,!7oo.

10. (ot/t o)«> 000 = ot (/-//,/) to.

11. (Of/)Go 00,700 =(0t/)^oo

12. (Of/oo) oo^oc =(0t/^oo) +

13. GO/GO oo^oo = -oo/yoo + oo

When the relative and non-relative operations occur
together, the rules of the calculus become pretty com
plicated. In these cases, as well as in such as involve
plural relations (subsisting between three or more ob
jects), it is often advantageous to recur to the numerical
coefficients mentioned on page 187. Any proposition
whatever is equivalent to saying that some complexus of
aggregates l and products of such numerical coefficients
is greater than zero. Thus,

^A>o

means that something is a lover of something ; and

JW<, > o

means that everything is a lover of something. We

1 The sums of page 188.

THE LOGIC OF RELATIVES. 201

shall, however, naturally omit, in writing the inequali
ties, the > 0 which terminates them all ; and the above
two propositions will appear as

The following are other examples : —

means that everything is at once a lover and a benefac
tor of something.

means that everything is a lover of a benefactor of itself.

means that there is something which stands to some
thing in the relation of loving everything except bene
factors of it.

Let a denote the triple relative " accuser to — of — ,"
and « the triple relative " excuser to — of — . Then,

means that an individual i can be found, such, that tak
ing any individual whatever, j, it will always be possible
so to select a third individual, k, that i is an accuser to
j of '&, and j an excuser to k of i.

Let TT denote " preferrer to — of — ." Then,

means that, having taken any individual i whatever, it
is always possible so to select two, j and k, that i is an
accuser to j of &, and also is either excused by j to & oi
ls something to which/ is preferred by k.

When we have a number of premises expressed in this
manner, the conclusion is readily deduced by the use of
the following simple rules. In the first place, we have

202 THE LOGIC OF EELATIYES.

In the second place, we have the formulae

In the third place, since the numerical coefficients are
all either zero or unity, the Boolian calculus is applicable
to them.

The following is one of the simplest possible examples.
Required to eliminate servant from these two premises :

First premise. There is somebody who accuses every
body to everybody, unless the unaccused is loved by
some person that is servant of all to whom he is not ac
cused.

Second premise. There are two persons, the first of
whom excuses everybody to everybody, unless the un-
excused be benefited by, without the person to whom he
is unexcused being a servant of, the second.

These premises may be written thus :

The second yields the immediate inference,

Combining this with the first, we have

2x2u2y2v(euyx + syvbvx) (axuv + syvlyu}.

Finally, applying the Boolian calculus, we deduce the
desired conclusion

Uyxaxuv + f«yJyu + «xuA-:r)-

The interpretation of this is that either there is some
body excused by a person to whom he accuses somebody,
or somebody excuses somebody to his (the excuser's)
lover, or somebody accuses his own benefactor.

THE LOGIC O*' RELATIVES. 203

The procedure may often be abbreviated by the use
of operations intermediate between II and £. Thus,
we may use Hr, II", etc. to mean the products for all
individuals except one, except two, etc. * Thus,

n/n/%+^

will mean that every person except one is a lover of
everybody except its benefactors, and at most two non-
benefactors. In the same manner, S7, £", etc. will de
note the sums of all products of two, of all products of
three, etc. Thus,

«'(W

will mean that there are at least three things in the
universe that are lovers of themselves. It is plain that
if m < n, we have

Um - IP 2n - ^'m.

(n/V) (n/%) -< np+'fai . yi)

Mr. Schlotel has written to the London Mathematical Society,
accusing me of having, in my Algebra of Logic, plagiarized from his
writings. He has also written to me to inform me that he has read
that Memoir with " heitere Ironie," and that Professor Drobisch, the
Berlin Academy, and I constitute a " lederliche Kleeblatt," with
many other things of the same sort. Up to the time of publishing
my Memoir, I had never seen any of Mr. Schlotel's writings ; I have
since procured his Logik, and he has been so obliging as to send me
two cuttings from his papers, thinking, apparently, that I might be
curious to see the passages that I had appropriated. But having ex
amined these productions, I find no thought in them that I ever did,
or ever should be likely to put forth as my own.

TIIE END.

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